



Copyright N?_ 

COPYRIGHT DEPOSIT. 














































% 















































































COPYRIGHT NOTICE 


This book was written by Mr. H. 
S. Jeffery expressly for the Interna¬ 
tional Brotherhood of Boilermakers, 
Iron Shipbuilders and Helpers of 
America. 

The book is the property of the 
Brotherhood, the contents first ap¬ 
pearing in the Brotherhood Journal 
each month during 1910. Each 
monthly article is protected by copy¬ 
right, and in E^ceh r b r 1910, the 
entire contents of this book were 
coyrighted. 


Copyright 1912. 

By International Brotherhood of Boiler Makers, 
Iron Shipbuilders and Helpers of America. 



BOILER CONSTRUCTION 
AND LAYING OUT 


BEING A TREATISE ON 


THE PROPERTIES OF BOILER MATERIALS, WORKSHOP 
PRACTICES, BOILER DESIGN AND CONSTRUCTION, 
AND LAYING OUT OF PATTERNS 


BY 

H. S. JEFFEEY 

\\ 

PRESIDENT OF THE JEFFERY CORRESPONDENCE SCHOOLS, FORMERLY 
PRINCIPAL OF THE SCHOOLS OF BOILERMAKING AND SHEET 
METAL PATTERN DRAFTING OF THE INTERNATIONAL 
CORRESPONDENCE SCHOOL. 



PEICE, $2.00 


A\ i °i i ^ 


fif 







T4 t 016 

.Ts 







/ /db? 2 * 

c 

# 

» 4 

c. r 
I « » 


5 $’ £,>00 

C Cl. A 312 4 5 3 

-?*# / 



PREFACE 


F OR the benefit of those who may contemplate making use 
of this work, wholly or in part, it is well to lay before 
them at the outset a general statement of the plan upon 
which it is written, together with some advice for the use and 
study of the same. 

Certain parts are theoretical in their nature, but aim to 
place before the reader all that is necessary to a thorough un¬ 
derstanding of the work, and, so to speak, present the practical 
side of theory. 

A statement of each part in prominent type appears at the 
head of eaeh demonstration, and as each figure is numbered, 
also lettered, an opportunity is afforded to trace each move 
easily and readily. 

The assumption is that the reader has no previous knowl¬ 
edge of the work. It is advisable in the study of all works of 
a scientific nature to begin at the beginning and take every¬ 
thing in its course. While each part will be complete in itself, 
some parts are necessarily carried farther into detail than others, 
and reference made from one part to another, pointing out simi¬ 
larity of principle. Individual solutions should not be sought; 
the principles or the foundation is the most important, for, when 
understood, no trouble should be experienced in applying them 
to work generally. 

It is possible that errors will creep in, and should any be 
discovered information to that effect will be cheerfully received. 

Yours very truly, 

H. S. JEFFERY. 






INDEX. 


xi 


INDEX 


BOILER CONSTRUCTION. 

A Page. 

Arrangement of Stays. 27 

Area, Computing the. 27 

Area Supported By a Rivet. 35 

Allowable Load On An Indirect Brace. 40 

B 

Boiler Material. 1 

Brass and Copper. 4 

Butt Joints. 10 

Brace to Plate, Relation of. 31 

Braces, Distribution of. 36 

Bracing Curved Surfaces. 40 

Boiler Capacity. 48 

Boilers, Horse Power of 50 

C 

Cold and Quench Test. 3 

Copper and Brass. 4 

Construction and Failure of Riveted Joints. 10 

Computing the Area . 27 

Collapsing Pressure of Tubes . 42 

Capacity, Boiler. 48 

D 

Distance From the Center of the Rivet Hole to the Edge of the Plate 16 

Distance Between Rows of Rivets. 17 

Designing a Rivet Joint, Practical Considerations.'. 25 

Distribution of Braces. 36 

E 

Extra Soft Steel and Rivet Steel. 2 

Efficiency of Riveted Joints. 10 

F 

Fire Box Steel. 2 

Flange or Boiler Steel. 2 

Factor of Safety. 24 

Furnace, Staying a. 40 

Feed-Water, Heating. 51 

H 

Heat and Steam. 48 
































xii INDEX. 

Page. 

Heating Surface. 49 

Horse Power of Boilers. 50 

Heating Feed-Water. 51 

I 

Iron and Cast Iron, Steel. 1 

Indirect Brace, Allowable Load. 40 

J 

Joints, Lap.•. 7 

Joints, Butt. 10 

L 

Lap Joints... 7 

M 

Materials, Boiler. 1 

O 

Open-Hearth Plate and Rivet Steel, Special. 1 

P 

Pitch of Rivets. 15 

Proportion of Parts of Rivet Joints. 17 

Practical Considerations When Designing a Riveted Joint. 25 

Proportion of Parts. 28 

Parts, Proportion of. 28 

Pitch of Stray Bolts. 30 

R 

Riveted Joints, Types of. 7 

Riveted Joints, Construction and Failure of. 10 

Riveted Joints, Efficiency of. 10 

Rivets, Shearing Strength of. 12 

Rivets, Pitch of. 15 

Rows of Rivets, Distance Between. 17 

Rivet Joints, Proportional Parts of. 17 

Rivet in Single Shear, Value of. 24 

Relation of Brace to Plate. 31 

Rivet, Area Supported by. 35 

Re-inforcing Rings . . ;. 46 

S 

Shearing Strength of Rivets. 12 

Shell, the Boiler. 4 

Shell, Boiler or Flange. 2 

Stresses. 1 

Steel, Iron and Cast Iron. 1 

Special Open-Hearth Plate and Rivet Steel. 1 

Soft Steel and Rivet Steel, Extra. 2 




































INDEX. 


xiii 


Page. 

Steel, Fire Box. 2 

Safety, Factor of. 27 

Staying of Surfaces. 27 

Staying, Arrangement of. 27 

Surfaces, Staying of. 27 

Stay Bolt, Pitch of. 30 

Supporting the Sheets. 38 

Sheets, the Support of. 38 

Surfaces. Bracing Curved. 40 

Staying a Furnace. 40 

Steam and Heat . .. 48 

Steam Making, Theory of. 48 

Surface Heating. 49 

Safety Valve. 51 

Separator. 52 

T 

Test, Cold and Quench. 3 

The Boiler Shell. 4 

Types of Riveted Joints. 7 

The Center of the Rivet Hole to the Eldge of the Plate, Distance from 16 

Tubes, Collapsing Pressure of. 42 

The Theory of Steam Making. 40 

V 

Value of Rivet in Single Shear. 24 

Valve, Safety. 51 


LAYING OUT. 

A 

A straight line . 108 

A perpendicular line. 108 

A vertical line. 108 

Acute angle. 108 

Altitude, the. 108 

Angle, right . 108 

Apex. 108 

Areas and circumference of circles. 120 

Avoirdupois weight . 180 

Angles or arc, measures of. 131 

B 

Base.108 

Ball. HI 

Bottom of threads, constant for finding diameter at. 116 






































xiv INDEX. 

Page. 

Boiling point in degrees . 119 

C 

Circumference, the . 56 

Curved line . 168 

Circle. 168 

Chord. 168 

Curve, irregular. 168 

Constant for finding diameter at bottom of thread. 116 

Circles, areas and circumference of. 120 

Cubic measure . 129 

D 

Drafting terms . 53 

Drawing, the perspective . 53 

Developing a square pipe cut off at an oblique angle. 57 

Developing a round pipe cut off at an oblique angle. 58 

Developing a square pipe fitting over the ridge of a roof. 59 

Developing a round pipe fitting over the ridge of a roof. 60 

Development of a 3-piece 90° elbow . 65 

Dome, laying out of . 65 

Developing a frustum . 96 

Developing an ellipse. 99 

Dodecagon. 114 

Diameter. 109 

Degrees. 109 

Decagon. 114 

Degrees, boiling point in . 119 

Dry measure . 130 

Decimals of a foot for each of an inch... ’. 132 

E 

Ellipse, developing an . 99 

F 

Frustum, developing a . 96 

G 

Globe. m 

Geometrical figures. 114 

H 

Hypotenuse. 113 

Hexagon. 114 

Heptagon. 114 

I 

Irregular curve . 109 




































INDEX. 


xv 


Page. 

L 

Laying out, principle of . 54 

Laying out a dome . 65 

Laying out a flaring transition piece. 67 

Lines, straight. 168 

Lines, parallel. 108 

Lines, curved. 166 

Lines, perpendicular. 168 

Lines, vertical . 108 

Linear measure . 129 

Liquid. 130 

Long ton table . 130 

M 

Measure, linear. 129 

Measure, square. 129 

Measure, cubic . 129 

Measure, liquid.:. 130 

Measure, dry. 130 

Measure of time . 131 

Measure of angles and arcs. 131 

Measures of money . 131 

Miscellaneous table. 131 

Minutes. 109 

N 

Nonagon. 114 

Number of gallons in round cisterns and tanks. 133 

Number of rivets in 100 pounds. 134 

O 

Offset. 109 

Octagon. 114 

Ogee curve. 109 

Obtuse angle. 109 

P 

Parallel lines, laying out by. 53 

Principles of laying out. 54 

Pentagon. 114 

Principles of triangulation. 73 

R 

Rolling, take-up in . 54 

Radius. 109 

Right angle . 110 

References. 135 






































xvi 


INDEX. 


Page. 

S 

Secant. Ill 

Sector. Ill 

Semi-circle . Ill 

Sphere. Ill 

Scalene triangle. 113 

Segment. 103 

Seconds. 103 

Square. 114 

Square measure . 123 

T 

Terms, drafting . 53 

The perspective drawing . 53 

Take-up in rolling . 54 

The joint between two pipes of different diameters intersecting at 

other than right angle . 63 

The development of a 3-piece 90° elbow. 65 

To develop a 2-piece 90° elbow. 69 

The patterns for a bifurcated pipe, the arms being the same diame¬ 
ter as the main pipe, and leaving it at the same angle. 71 

Triangulation, principle of . 73 

To develop the patterns for a transition piece between a rectangular 

opening and a round pipe. 83 

To develop the pattern for a taper course or slope course. 86 

To develop the pattern for a four-way branch Y. 88 

To develop the pattern for a cone. 94 

To develop the pattern for a foot-tub. 97 

To develop the pattern for a telescoping transition piece intersect¬ 
ing a cylinder . 101 

To develop the pattern of a telescoping transition piece intersecting 

a cylinder at an angle. 102 

at an angle . 102 

To develop the pattern of a transition piece having both the upper 

and the lower bases obliquely inclined. 106 

Tangent. 110 

Triangle. 114 

Table of decimals equivalents . 115 

Time, measure of . 131 

Troy weight . 130 

U 

ITndecagon. 114 

V 

Vertical line. 108 

Vertex. 110 






































BOILER CONSTRUCTION. 


1 


BOILER CONSTRUCTION 


BOILER MATERIALS 

STRESSES 

STEEL, IRON AND CAST IRON. 

1. The materials used in boiler construction are steel, iron and 
cast iron, also copper, brass and various alloys. Steel has supplanted 
iron, and because it can be made as cheap, if not cheaper-and because 
of a greater tensile strength. Also is homogeneous, whereas iron is not. 

The term homogeneous means that all parts are in equation - similar 
in material-or of same composition or structure throughout. An iron 
plate will stand a greater stress with the grain than against the grain. 
It therefore is not in equation and not homogeneus. 

2. Steel is made from iron, and the latter is chiefly derived from 
ores, as bagnitite, hematite and imonite, which are abundant and widely 
distributed. As found in commerce it is never pure, but is combined 
with small quantities of carbon, phosphorus, silicon, etc. The com¬ 
mercial steel is made by either the Bessemer or the open-hearth process. 
The former was patented in 1855 by Henry Bessemer. The process 
consisted of eliminating the carbon and silicon from the pig iron pre¬ 
paratory to its conversion into steel or ingot iron, by forcing a blast 
of air through the metal while melting. 

3. The open-hearth, or Siemens Martin process, is the making 
of steel in which the pig iron is decarbonized by melting in an open- 
hearth regenerative furnace in combination with scrap iron and iron 
ore. The Association of American Steel Manufacturers have adopted 
the following standard specifications for open-heart plate: 

SPECIAL OPEN-HEARTH PLATE AND RIVET STEEL. 

4. Steel shall be of three grades: Extra soft, fire-box and flange 
or boiler. 



2 


BOILER CONSTRUCTION. 


EXTRA SOFT STEEL AND RIVET STEEL. 

Ultimate strength, 45,000 to 55,000 pounds per square inch; elastic 
limit, not less than one-half the ultimate strength; elongation 26 per 
cent; cold and quench tests, 180 degrees flat on itself; without frac¬ 
ture on hent portion; maximum phosphorus, .04 per cent, and maxi¬ 
mum sulphur, .04 per cent. 

FIRE BOX STEEL. 

Ultimate strength, 52,000 to 62,000 pounds per square inch; elastic 
limit, not less than one-half the ultimate strength; elongation, 26 per 
cent; cold and quench tests, 180 degrees flat on itself, without fracture 
on bent portion; maximum phosphorus, .04 per cent; and maximum 
sulphur, .04 per cent. 

FLANGE OR BOILER STEEL. 

Ultimate strength 55,000 to 65,000 pounds per square inch; elastic 
limit not less than one-half the ultimate strength; elognation 26 per 
cent; cold and quench tests 180 degrees flat on itself, without frac¬ 
ture on bent portion; maximum phosphorus .06 per cent, and maxi¬ 
mum sulphur .04 per cent. 

5. The ultimate or tensile strength means the ability to resist a 
given tension or stress. When a force acts it tends to draw apart 
the parts of a body, especially of a line, cord or sheet, combined with 
an equal and opposite system of resisting forces of cohesion holding 
the parts of the body together. Thus, when an ultimate or tensile 
strength of 60,000 pounds is stated, it means that one square inch 
of solid metal will approximately resist a force of 60,000 pounds before 
pulling apart. 

6. The ELASTIC LIMIT, or the LIMIT OF ELASTICITY, is the 
point of stress beyond which an elastic body loses power to return com¬ 
pletely to its former shape and size. If the load applied exceeds the elas¬ 
tic limit, then the plate is stretched, or lengthened - and no boiler or struc¬ 
ture should be subjected to a stress equal to, or beyond the elastic limit. 

The DUCTILITY of a plate means that it is capable of being drawn 
out. In Fig. 1 is shown the test piece as advocated by the Association 
of American Steel Manufacturers, and the following are the rules gov¬ 
erning the test: 

7. All tests and inspections shall be made at place of manufac¬ 
ture prior to shipment. The tensile strength, limit of elasticity and 
ductility shall be determined from a standard test piece, cut from the 
finished material, the standard shape of said test piece for sheared 
plates to be as shown in Fig. 1. Test coupons cut from other material 
than plates may be the same as those for the plates, or they may be 


BOILER CONSTRUCTION. 


3 


planed or turned parallel throughout their entire length. The elonga¬ 
tion shall be measured on an original length of eight inches, except 
in round of five-eighth inch or less in diameter, in which case the 
elongation shall he measured in the length, equal to eight times the 
diameter of the section tested. Four coupons shall be taken from each 
melt of finished material; two for tension and two for bending. 

Material, which is to be used without annealing or further treat¬ 
ment, is to be tested in the condition in which it comes from the rolls. 

TEST PIECE. 


rxo3’'«/so 


1/1 

MOT CESS THAN 3' 

ttl'"}" 

J 

iLfXfJ ETC 

-ABOUT »©* -- 


”1 

ABOUT £L“ 


Fig. 1. 


When material is to be annealed, or otherwise treated before use, the 
specimen representing such material is to be similarly treated before 
testing. Each finished piece of steel shall be stamped with the melt 
number. All plates shall be free from surface defects, and to have a 
workmanlike finish. 


COLD AND QUENCH TEST. 

8. The COLD test is merely bending a section of plate in its nat¬ 
ural condition, and the QUENCH test is the heating and then the cool¬ 
ing of the plate, which tends to harden it-and then the bending 
operation in the manner set forth in the specifications of the Associa¬ 
tion of American Steel Manufacturers. 

9. Cast iron, which is a commercial iron produced in a blast fur¬ 
nace and containing a large proportion of carbon, some of which is 
segregated, is neither ductile nor malleable, but brittle. In addition 
it is liable to have more or less blow-holes. It is fast passing away 
in boiler construction, and some states and cities prohibit its use in 
boiler construction, except such parts not subject to pressure. 

10. Wrought iron is a commercial iron produced by a puddling 
furnace or forge, and contains little carbon and other substances. It 
usually is fibrous, ductile and malleable. Bar-iron, weld-iron and steel 
are various differing compounds of iron, containing less carbon than 
















4 


BOILER CONSTRUCTION. 


cast iron and more than wrought iron that can be forged, tempered 
and sensibly hardened by heating to redness and suddenly cooling. 
The three varieties of manufactured iron differ not only in the degree 
of their properties, but also in the proportions of their constituents. 

Malleable iron is cast iron that has been rendered tough and malle¬ 
able by long continued high heating while embedded in powdered hema¬ 
tite, ferric oxid or some other decarbonizing material, and allowed to 
cool slowly. 

COPPER AND BRASS. 

12. A reddish ductile metallic element; called copper, was form¬ 
erly used extensively in boiler construction for firebox furnace sheets, 
staybolts, and to some extent for flues. The cost of copper, however, 
is so great that it is not used to any extent for the foregoing pur¬ 
poses in the United States of America, though it is used more or less 
in boiler construction in some of the foreign countries. An alloy of 
copper and zinc, called brass, which is harder than copper and quite 
ductile, is used for washout plugs and various fittings. Formerly an 
alloy of copper, especially with tin, was called brass. The modern alloy 
of zinc came into use in the last century. 

THE BOILER SHELL. 

13. A boiler shell or cylinder subject to internal pressure has a 
force that tends to rupture it through the longitudinal plane, or length¬ 
wise. And a force that tends to rupture it transversally, or cross¬ 
wise. In Fig. 2 the arrows A and B represent the force that tends 
to part the cylinder transversally, and the arrows C and D represent 
the force that tends to part the cylinder longitudinally. 

14. The latter force is the one of greatest concern-and, for the 
reason that the total force acting on the cylinder transversally is ap- 



Fig. 2. 


proximately one-half as great as the total force acting on the cylin¬ 
der longitudinally, or in other words, the force acting on one square 







BOILER CONSTRUCTION. 


5 


inch of solid plate in the longitudinal plane, therefore, the transverse 
strength, considering only a seamless cylinder, is twice as strong as 
the longitudinal strength. 

15. The total force tending to part the cylinder, Fig. 2, through 
the longitudinal plane is equal to the working pressure per square 
inch multiplied by the cross-sectional area of the cylinder. Assum¬ 
ing the inside diameter of Fig. 2 to be 66 inches and the length 14 
feet, the cross-sectional area is equal to the length of the cylinder in 
inches multiplied by the inside diameter in inches-and in the fore¬ 
going case will be: 168 (length in inches) x 66 (diameter in inches) 
= 11,088 square inches. If the working pressure is 100 pounds per square 
inch, the total force acting on the longitudinal plane will be: 100 
(working pressure) x 11,088 (surface exposed to pressure) = 1,108,800 
pounds. 

16. The force to resist the foregoing will depend upon the ten¬ 
sile strength and the thickness of plate. If a five-eighths inch plate, 
having a tensile strength of 55,000 pounds, is used the resisting force 
will be the length of the plate in inches multiplied by the thickness 
in inches, multiplied by the tensile strength; or, in the above instance, 
will be: 168 (length in inches) x| (thickness in inches) x2 (both 
sides) x 55,000 (tensile strength) = 6,930,000 pounds. The calculations 
bring out that the force acting longitudinally on the cylinder is 1,108,* 
800 pounds, while the resisting force is 6,930,000 pounds; there¬ 
fore the ration, or the factor of safety, between the bursting pressure 
and the working pressure, is: 6,930,000 -f- 1,108,800 = 6.25. 

17. The force acting transversally on the cylinder is equal to its 
cross-sectional area multiplied by the working pressure per square inch, 
which, in this case, is 66 x 66 x 7,854 x 100 = 342,100 pounds. The force 
to resist the foregoing is the strength of the solid plate in the trans¬ 
verse plane. The area of the plate is found by multiplying the cir¬ 
cumference by the neutral diameter, which is equal to the inside diame¬ 
ter, plus one thickness of plate, or 66| (neutral dimaeter) x 3.1416 (con¬ 
stant) x! (thickness) = 78.2 square inches. The resisting force is then 
78.2x55,000 = 4,301,000 pounds. 

The calculations bring out that the force acting on the transverse 
plane is 342,100 pounds and the resisting force is 4,301,000 pounds, 
therefore, the bursting pressure exceeds the working pressure by 4,301,- 
000 divided by 342,100 = 12.5, thus showing that the transverse load is 
approximately one-half the longtitudinal load, and this is true regard¬ 
less of the pressure per square inch. 

18. The foregoing calculations are based on a seamless, or joint¬ 
less cylinder. A large boiler shell, however, is not seamless, unless 
welded, and as a welded joint is not to be depended upon, practically 
every boiler, tank and similar structure is constructed with a riveted 
joint. The method of connecting the plates necessitates rivet holes, 


6 


BOILER CONSTRUCTION. 


and the punching or the drilling of the holes in the plate injuries the 
plate—that is to say leaves less solid plate to resist the forces acting 
upon the respective planes. 

Due reflection over these statements will show that if the longitu¬ 
dinal joint, regardless of the type, could be made as strong as the 
solid plate, which is 100 per cent, the fact that the transverse ratio 
or resisting power to the working pressure is twice as great as the 
longitudinal ratio or resisting power to the working pressure per¬ 
mits the transverse point-called the GIRTH SEAM-to be made one- 
half the strength of the longitudinal seam-and if this was done the 
ratio between the bursting and the working pressure would, theoretically, 
be in equation throughout the cylinder. 

19. In boiler and tank construction, etc., the aim should be to 
make the cylinder as round as practicable, for if otherwise the inter¬ 
nal pressure causes the cylinder, unless braced, to form to a circle. 
If the cylinder is round in the first place the pressures causes no 
rounding up, but merely maintains the round form, but if the cylin¬ 
der is not round when the pressure is applied it undergoes a change 
in form, and when the pressure is relieved the cylinder returns to 
Us old form. The constant changing of the shape works the metal 
and also causes the riveted seams to leak. Since such undue stresses 
can not be calculated, the assumption in the case is that the cylinder 
has been made as round as practicable. 

20. The factor of safety is merely the ratio between the working 
pressure and the bursting pressure. The factor of safety, however, 
is not the ratio based on the longitudinal seam per 100 per cent basis, 
but upon the actual percentage of the joint and the percentage of 
the joint is called the EFFICIENCY. In the calculation, Art. 16, the fac¬ 
tor of safety for the longitudinal seam, which was taken at 100 per cent, 
~as found to be 6f, but if the efficiency of the longitudinal joint should 
be 75 per cent, then the factor of safety would be: .75x61 = 4.54 factor 
of safety. 

If 100 pounds pressure must be maintained and a factor of 5 is 
desired, which is the generally adopted factor of safety for commer- 
ceial work, the efficiency must be made greater than 75 per cent, or 
the thickness of the plate used be heavier than three-eighth inch. All 
the foregoing shows that the allowable working pressure depends upon 
the thickness of the plate, upon the diameter of the boiler; the factor 
of safety; the efficiency of the joint and the tensile strength of the 
plate. The allowable working pressure may be readily found by the 
following formula: 

21. Where: 

TS = Tensile strength of plate in pounds. 

T = Thickness of plate in inches. 

D = Diameter of boiler in inches. 


BOILER CONSTRUCTION. 


7 


F = Factor of safety. 

E = Efficiency of joint. 

A = Allowable working pressure per square inch. 

TS x T x 2 x E 

-= A 

DxF 

Considering the 66-inch by 14-foot boiler with i-inoh plate, ten¬ 
sile strength 55,000 pounds, the efficiency of the longitudinal seam as 
80 per cent, and the factor of safety as 5, the allowable working pres¬ 
sure by employing the formula will be 

55,000 x.375 x 2 x.80 

-= 100 pounds working pressure. 

66x5 

22. When testing the boiler care should be exercised not to apply 
an excessive pressure, for though a factor of 5 is used, the elastic 
limit of the plate, as mentioned in Art 4, is only 50 per cent of the 
ultimate strength, and this being true limits the pressure for testing 
purposes. Merely because a boiler stands a pressure far beyond the 
working pressure desired is no indication that it is safe. Theoretically, 
if the boiler in the foregoing instance is subjected to a cold water or 
hydrostatic test of 275 pounds (which it might stand without a leak), 
it has been injured rather than benefitted, as the test applied reached 
the elastic limit- that point where the plate is stretched and does not 
return to its former shape and size. The placing of the holes in the 
sheet by punching may injure the plate more or less, and though, 
theoretically, the elastic limit may be 50 per cent of the ULTIMATE 
strength, it may, due to the foregoing, be considerably less, therefore, 
if the hydrostatic test is made one and one-half times the working 
pressure it will serve as much good, if not more good, than if a greater 
pressure is applied. 


RIVETED JOINTS; THEIR CON¬ 
STRUCTION AND FAILURE 

TYPES OF RIVETED JOINTS 

LAP JOINTS. 

23. In Fig. 3 is shown two plates riveted with one row of rivets. 
When plates are so riveted the joint is called a SINGLE-RIVETED LAP 
JOINT. When two rows of rivets are used, as shown in Fig. 4, the joint 
is called a DOUBLE-RIVETED LAP JOINT, and when three rows of rivets 
are used, as shown in Fig. 5, the joint is called a TRIPLE-RIVETED LAP 
JOINT. 






8 


BOILER CONSTRUCTION. 






































































































BOILER CONSTRUCTION. 


9 








Fig. 7. 
































































































10 


BOILER CONSTRUCTION. 


BUTT JOINTS. 

24. When plates are secured together with straps called BUTT 
STRAPS, also WELT STRAPS, as shown in Fig. 6, the joint is called a 
SINGLE-RIVETED DOUBLE-STRAPPED BUTT JOINT. It is single 
riveted for the reason that the severing of one row of rivets severs the 
joint-that is, the plates heretofore fastened together can he readily sep¬ 
arated. If only one butt strap is used, either inside or outside, the joint 
is a SINGLE-RIVETED SINGLE-STRAP JOINT, but this form of riveted 
joint is not extensively used. 

In Figs. 7 and 8 two forms of riveted joints are shown-both are 
called DOUBLE-RIVETED DOUBLE-STRAPPED BUTT JOINTS. When 
the riveted joint, Fig. 8, is meant it is usually designated by stating the 
inner strap extended. In Figs. 9 and 10 two types of riveted joints are 
shown. Both are called TRIPLE-RIVETED DOUBLE-STRAPPED BUTT 
JOINTS. The construction is similar to the joints Figs. 7 and 8, except 
one additional row of rivets on each side of the butt joint is added. In 
Fig. 11 is shown a riveted joint, called a DOUBLE-STRAPPED QUAD- 
RUPLED-RIVETED BUTT JOINT. It is constructed so that the inner 
strap extends sufficiently to permit four rows of rivets on each side of 
the butt joint, but only two rows of rivets are in both the inner and 
outer straps. The two outside rows of rivets, as shown in Fig. 11, fasten 
the inner strap to the shell only. This type of joint must be made as 
nearly equal to the strength of the solid plate as possible. There are 
many types of riveted joints, but those shown, Figs. 3 to 11 inclusive, 
are the types of riveted joints generally employed. 

CONSTRUCTION AND FAILURE OF RIVETED JOINTS. 

EFFICIENCY. 

25. The EFFICIENCY of a riveted joint is the lowest per cent of its 
several parts. The pitch of the rivets, which is the distance from 
the center to the center of the rivets, is 100 per cent. It matters not 
the pitch, the percentage is 100. When holes are installed into the 
plate a certain amount of plate is removed, therefore the resistance 
that is to resist the stress acting on the shell, is the resistance offered 
by the net sections of plate left between the rivet holes. 

26. If a plate one inch thick has rivets, 4-inch pitch, the 4-inch 
pitch representes 100 per cent. The area of the plate in this case is: 
4x1 = 4 square inches. If the tensile strength is 60,000 pounds, then 
the 4 square inches has a resisting force of 4 x 60,000 = 240,000 pounds. 
Were holes one inch in diameter installed into the plate the distance 
from the edge of one rivet hole to the adjoining rivet hole would be 
4-1 = 3 inches. It will be seen that one inch of metal having been 
removed the resisting force is reduced. 

The section of plate between the rivet holes is called the NET SEC- 


BOILER CONSTRUCTION. 


11 


TION PLATE, and if the rivet holes are laid off uniform, as they should 
be, each and every section will be alike, therefore what is the efficiency 
for one section is the efficiency for all like sections. The efficiency 
of the net section of plate is not decided by its thickness, or by its 
tensile strength, but by the pitch and the size of the rivet hole. With 
a 4-inch pitch and 1-inch rivets, the efficiency of the net section of 
plate may be found by the following formula: 

Where: 

P = Pitch of rivets in inches from centers to center. 

D = Diameter of rivet holes in inches. 

E = Efficiency. 

P-D (4-1) x 100 

-= E, or-= 75 per cent. 

P 4 

27. Assuming the pitch to be 3-inch, and 1-inch rivet holes, then 
the efficiency will be: 

P-D (3-1) x 100 

-= E, or-= 66.66 per cent. 

P 3 



As will be seen it amounts to subtracting from the pitch the 
diameter of the rivet and dividing the product by the pitch. The ex¬ 
amples further show that the greater the pitch the greater the length 
































































12 


BOILER CONSTRUCTION. 


of the net section of plate; hence more resistance, also greater effi¬ 
ciency. Further still, the examples show that the larger the diameter 
of the rivet the more metal removed, and a corresponding reduction in 
the resistance of the net section of plate, also efficiency. 



28. A stress acting on a riveted joint does not act solely upon 
the net sections of plate. It also acts upon the rivets. The strength 
of the rivets compared to the plate must be ascertained. If weaker 
than the net section of plate, then the efficiency of the joint is decided 
hy the strength of the rivets. If stronger than the net section of 
plate, then the efficiency of the latter is the efficiency of the joint. 

SHEARING STRENGTH OF RIVETS. 

29. The shearing strength of a rivet does not depend solely upon 
the metal of which the rivet is composed and its size. The manner 





























































































BOILER CONSTRUCTION. 


13 


in which the rivet is installed is very important. If the rivet is in¬ 
stalled in a single riveted lap joint, as shown in Fig. 3, then the rivet 
is in single shear. The expression in SINGLE SHEAR means severing 
the rivet, as shown in Fig. 12. The shearing strength of a rivet is 
found hy multiplying its area by the shearing strength per square inch. 
The expression in DOUBLE SHEAR means the severing of the rivet at 
two places at one and the same time, as shown in Fig. 13. The rivets in 
a single-riveted butt joint, as shown in Fig. 4, are in double shear. 

30. Notwithstanding that a rivet in double shear has twice as 
great an area as a rivet in single shear, tests have brought out that 
a rivet in double shear will shear when subject to a force not quite 
twice as great as the force required to shear a rivet in single shear. 
Generally figuring the force required to shear a rivet in double shear 
is 1.85 times the force' required to shear a rivet in single shear. 



Fig. 10. 


31. However, the Master Steam Boiler Makers’ Association, now 
the International Master Boiler Makers’ Association, conducted a number 
of tests, and in 1906 moved, seconded and unanimously carried to make 
the following recommendations: 

Iron rivets in single shear, 42,000 pounds shearing strength per 
square inch. 































































































14 


BOILER CONSTRUCTION. 


Steel rivets in single shear, 45,000 pounds shearing strength per 
square inch. 

Iron rivets in double shear, 80,000 pounds shearing strength per 
square inch. 

Steel rivets in double shear, 88,000 pounds shearing strength per 
square inch. 



Fig. 11. 


If steel rivets were used in the riveted joint, Art. 26, the efficiency 
will he the strength of one rivet divided by the strength of the plate, or 

lx lx.7854x45,000 x 100 

-= 14.7 per cent 

4 x 60,000 

32. The efficiency of the plate in the example, Art. 26, is 75 per 
cent, while the rivet efficiency is found to be only 14.7 per cent, there¬ 
fore the efficiency of the joint is the latter efficiency. To increase the 
efficiency of the rivet one or more of the following must be done: Re¬ 
duce the pitch of the rivets; use a larger rivet; or use a lighter plate. 




























































BOILER CONSTRUCTION. 


15 


To reduce the pitch increases the rivet efficiency, but reduces the effi¬ 
ciency of the net section of plate. To use a larger size rivet increases 
the rivet efficiency and reduces the efficiency of the net section of plate 
-there being no change in the rivet pitch. To use a lighter plate, 
without change in either the rivet pitch, or change in the size of the 
rivet, does not alter the plate efficiency, but increases the rivet effi¬ 
ciency and reduces the allowable working pressure, and because a re¬ 
duction in the thickness of the boiler shell means a reduction in the 
resistance of the net section of plate. 

33. In Art. 27 the pitch is given as 3 inches, and the efficiency 
of the rivets, using steel rivets, is: 

lx lx. 7854x45,000 X 100 

--= 19.6 per cent 

3 x 60,000 

The foregoing examples set forth clearly that a 1-inch rivet is too 
small with 4-inch rivet pitch. Aside from the steam-tight joint, the 
joint is not allowable due to the low efficiency of the rivets. Reduc¬ 
ing the pitch from 4-inch to 3-inch makes only a slight increase in 
the rivet efficiency; hence the remedy lies in employing additional 
rows of rivets. 

Maintaining the 3-inch pitch, which as set forth in Art. 27, gives 
a plate efficiency of 66.66 per cent, the addition of another row of rivets, 
thus making a double-riveted lap joint, increases the shearing resist¬ 
ance of the rivets twice, or makes the efficiency 2 x 19.6 = 39.2 per cent. 
By adding another row of rivets, thus making the joint a triple-riveted 
joint, the rivet efficiency is further increased and is: 3x19.6 = 58.6 per 
cent. 

The calculations, however, show that the plate efficiency is greater 
than the rivet efficiency, thus another row of rivets must be added, 
or a larger size rivet used, or the plate reduced in thickness, or the 
pitch reduced. In designing a riveted joint the aim at all times should 
be to make the net section of plate the weakest part of the riveted 
joint. 

PITCH OF RIVETS. 

34. The pitch of the rivets can not be decided solely from one 
source. The pitch must not be excessive so that difficulty will be ex¬ 
perienced in keeping the seam and rivets steam-tight. The pitch de¬ 
pends upon the size of the rivet; thickness of plate and type of riveted 
joint. The more rows of rivets in a riveted joint the greater the pitch 
allowable. However, there are approximate pitches, but there is no 
set standard, or any hard and fast rule governing the pitch of rivets. 

The old time-honored rule of making the diameter of the rivet about 
twice as great as the thickness of plate, works well enough in places, 



16 


BOILER CONSTRUCTION. 


but with heavy plate such a rule is worthless. No one would think 
of using such a rule if the shell plate was 1 y 2 inches thick. Other 
rules similar in nature to decide the pitch are unreliable. The cor¬ 
rect way is to figure out each part and construct the joint so that its 
weakest part is the net section of plate, maximum pitch. Trial pitches 
may, however, be found by the following: 


Where: 

T = Thickness of plate in inches. 

P = Maximum pitch of rivets. 

C = Constant applicable from table. 


TABLE I. 

Constants for 

Rows of Constants for double-strapped 

rivets lap joints butt joints 


1 

1.31 

1.75 

2 

2.62 

3.50 

3 

3.74 

4.63 

4 

4.14 

5.52 


(CxT) +lg = P. 


With a single-riveted lap joint as shown in Fig. 3, the maximum 
pitch with a maximum pitch with a i-inch plate would be: 1.31 (con¬ 
stant) x.375 (thickness of plate in inches - expressed decimally) equals 
.49125. To this add 1§, or as expressed decimally, 1,625, making the 
maximum pitch 2.11625 inches, which is about 2i inches. 

The constants for double-strapped butt joints are intended for only 
the pitch of rivets in double shear. For instance the constant for the 
triple-riveted double double strapped butt joint (Fig. 9) is 4.63 while the 
constant for the triple-riveted double-strapped butt joint (Fig. 10) is 
3.50. 


DISTANCE FROM THE CENTER OF THE RIVET HOLE TO THE 

EDGE OF THE PLATE. 

35. The distance from the center of the rivet hole to the edge of 
the plate is called the LAP, though technically the expression erro¬ 
neous. The lap proper is twice the distance, a, (Fig. 3). To make no 
departure from shop terms, all reference to the lap will mean the dis¬ 
tance from the center of the rivet to the edge of the plate. 

Though there are a vast number of rules, etc., for figuring out 
the lap, tests have been brought out that making the lap a, (Figs. 3 
to 11 inclusive), 1J times the diameter of the rivet hole is sufficient 
to prevent the plate from either shearing or crushing in front of the 
rivet; hence, all calculations to the amount of the lap amounts to prac¬ 
tically nothing. However, the exception is the riveted joint (Figs. 
6 and 8). The distance b should be made at least 11 times, preferably 



BOILER CONSTRUCTION. 


17 


2 times, the diameter of the rivet hole. The distance & (Figs. 7, 9, 
10 and 11), should never be less than 11 times, preferably 11 times, 
the diameter of the rivet hole. 

DISTANCE BETWEEN ROWS OF RIVETS. 

36. The distance between rows of rivets of a riveted joint de¬ 
pends upon its construction. The distance should not be so excessive 
as to permit the plate to spring between the rivets. On the other hand 
the distance should not he so small that the rivet set, or rivet die, 
when driving a rivet, will come in contact with and disfigure an ad¬ 
joining rivet previously driven. 

The distance between the rows of rivets as designated by the let¬ 
ter c (Figs. 4, 5, 7, 9, 10 and 11), should not exceed three times the 
diameter of the rivet hole, and should not be less than 2 times the 
diameter of the rivet hole. In the double-strapped butt riveted joint 
with inner strap extended, as shown in Figs. 8, 10 and 11, the dis¬ 
tance d should be sufficient to not only permit the rivets being readily 



Fig. 12. 



Fig. 13. 


driven, but also to permit the outer butt strap to be easily calked at 
all points. The distance is rarely made greater than 4 times the diame¬ 
ter of the rivet hole, or less than 3 times the diameter of the rivet 
hole. In the quadruple butt joint, double-strapped and with inner strap 
extended, the distance e (Fig. 11), should be made not less than, 21 
times, and not to exceed 31 times, the diameter of the rivet hole. 


























































































18 


BOILER CONSTRUCTION. 


PROPORTIONS OF PARTS OF RIVETED JOINTS. 

37. To attempt to design a riveted joint, particularly a double- 
strapped butt joint, so that all parts are about equal, is out of all 
question, but with lap joints, single, double and triple-riveted, the effi¬ 
ciency of the net section of plate and efficiency of the rivets will often 
be nearly alike - sometimes varying only 1 per cent. 

However, with a double-strapped butt joint many of the rivets are 
in double shear - in some double-strapped butt joints all the rivets are 
in double shear-see Figs. 7 and 9-and the rivet efficiency of such 
joints is generally many per cent greater than the efficiency of the 
net section of plate. 

The rivets in the riveted joints, as shown in Figs. 8 and 10, are 
part in single shear and part in double shear. This requires that the 
efficiency of the set section of plate between the inner row of rivets 
and the efficiency of the rivet in single shear be added together to 
ascertain if their strength is as great as the strength of the net sec¬ 
tion of plate, maximum pitch of rivets. 



The foregoing is clearly brought out in the triple-riveted joint, as 
shown in Fig. 14. The following is the specification for the riveted 
joint: Maximum pitch 7J inches; minimum pitch 31 inches; diameter 








































































BOILER CONSTRUCTION. 


19 


Fig. 15. 



of rivet hole 1 inch; thickness of plate \ inch; tensile strength of plate 
60,000 pounds, and shearing strength of steel rivets, 45,000 pounds 



Fig. 16. 



















































































20 


BOILER CONSTRUCTION. 


per square inch for rivets in single shear, and 88,000 pounds per square 
inch for rivets in double shear. 

Since the maximum pitch of rivets is H inches and the diameter 
of the rivet hole is 1 inch, the length of the net section of plate will 
be 7i-l = 6i inches. The efficiency is: 

6.5x100 

-= 86.6 efficiency 

7.5 

38. To determine if the net section of plate, maximum pitch, is 
the weaker of all parts, requires that the other parts be computed. 
Reference to Fig. 14 shows that the pitch of rivets of the inner rows 
is 31 inches. The efficiency of the net section of plate of the inner 
rows may be computed two ways, as follows: 

First: Subtract from the maximum pitch (in this case 7i inches) 
the diameter of two rivet holes, or 7|-2 = 5i inches. Then the effi¬ 
ciency is: 

5.5x100 

-= 73.3 efficiency 

7.5 

Second: Subtract from the minimum pitch (in this case 31 inches) 
the diameter of one rivet hole, or 31 -1 = 21 inches. Then the effi¬ 
ciency is: 

2.75 x 100 

-= 73.3 efficiency 

3.75 

39. For the riveted joint to fail by rupture through the net sec¬ 
tion of plate of the inner row of rivets, the rivet in the outer row 
must be sheared, therefore its efficiency is to be added to the efficiency 
of the net section of plate, minimum pitch. The efficiency of the rivet 
may be found by the formula. 

Where: 

N = Number of rivets in shear. 

TS = Tensile strength of plate in pounds per square inch. 

T = Thickness of plate in inches (express decimally). 

P = Pitch of rivets. 

SS = Shearing strength of rivest in pounds per square inch. 

A = Area of one rivet. 

E = Efficiency. 

Ax SS x N 

-= E, 

TSxTxP 

.7854x45,000x1x100 


60,000 x .5 x 7.5 


= 15.7 efficiency. 







BOILER CONSTRUCTION. 


21 



Fig. 17. 



Fig. 18. 


















































































22 


BOILER CONSTRUCTION. 


40. Thus 73.3 +15.7 = 89 efficiency, which exceeds by over 2 per 
cent the efficiency of the net section of plate, maximum pitch. Had 
iron rivets, which, according to Art. 31, have a shearing strength of 
42,000 pounds per square inch, been used in place of steel rivets, then 
the efficiency of the rivet in single shear would be less, and because 
its shearing strength is less. Assuming iron rivets were used, the 
efficiency would be: 

42,000 x.7854x100 

-= 74.6 efficiency 

60,000 x .5 x 7 x 7.5 

Thus 73.3 + 14.6 = 87.9 efficiency - say 88 efficiency, which is also 
greater than the efficiency of the net section of plate maximum pitch. 
The foregoing calculations show (in this case) that with either iron or 
steel rivets the combined efficiency of the net section of plate, mini¬ 
mum pitch, and the efficiency of the rivet in single shear, is greater 
than the efficiency of the net section of plate, maximum pitch. It 
further brings out that a change in the shearing strength of 3,000 
pounds does, in this case, reduce the efficiency 1 per cent. Had the 
combined efficiency of the net section of plate and rivet in single 
shear only exceeded, using steel rivets, the efficiency of the net sec¬ 
tion of plate, maximum pitch, by less than 1 per cent, it will be seen 
that by using iron rivets the combined efficiency would be less than 
the efficiency of the net section of plate, maximum pitch, and this 
being true, the efficiency of the joint insofar as calculated at present 
would be the combined efficiency of the two parts. As the calcula¬ 
tion, both with iron and steel rivets, shows that the combined efficiency 
exceeds the efficiency of the net section of plate, maximum pitch, the 
efficiency of the latter is the efficiency of the riveted joint, insofar 
as calculated. 

41. To determine if the riveted joint may fall by shearing all 
the rivets, their efficiency, both the rivets in single shear and double 
shear, must be ascertained. Inspection of Fig. 14 shows that within 
the maximum pitch there are four rivets in double shear and one rivet 
in single shear. The efficiency with steel rivets in double shear is: 

4 x.7854x88,000x100 

-= 122.87 efficiency 

60,000 x .5 x 7.5 

As will be seen the efficiency of the rivets in double shear is con¬ 
siderable more than the efficiency of the net section of plate, maxi¬ 
mum pitch. To the 122.87 efficiency is to be added the efficiency of 
the rivet in single shear which, according to Art. 39, is 15.7, thus mak¬ 
ing the total efficiency to 122.87 +15.7, or 138.57 efficiency. 

42. A remote mode of failure of the riveted joint is for the butt 
straps, both inside and outside straps, to fracture through the net sec- 




BOILER CONSTRUCTION. 


23 




Fig. 20. 















































































































































24 


BOILER CONSTRUCTION. 


tion of plate, minimum pitch. If the least thickness of the butt straps 
is made equal to three-fourths of the thickness of the shell plate, the 
thickness is sufficient as far as strength is concerned. However, with 
light shell plates the outer strap should be as thick, and in some cases 
thicker, than the shell plate, and for the reason that a light butt strap 
is liable to spring between the rivets unless calked very carefully. 
Some boiler manufacturers, regardless of the thickness of the shell 
plate, make the outer strap the same thickness. 

VALUE OF RIVET IN SINGLE SHEAR. 

43. The value of the rivet in single shear of the riveted joint 
(Fig. 14) is apparent, for, were the outer row of rivets to be omitted 
the joint would then be a double-strapped joint (Fig. 7). According 
to the calculations of the net section of plate (Fig. 14) as given in 
Article 38, its efficiency is 73.3, therefore the rivet is single shear and 
has a large bearing on the efficiency. 

If the dropping of the outer row of rivets reduces the efficiency sev¬ 
eral per cent then the adding of another row should tend to increase 
the efficiency; hence, the double-strapped quadruple-riveted butt joint, 
as shown in Fig. 15. As will be seen Fig. 15 is the riveted joint, Fig. 
14, with an additional row of rivets in single shear. The rivets are 
arranged so that the maximum pitch is 15 inches, creating a greater 
length of net section of plate. The efficiency is: 

(15-1) x 100 

-= 93.3 efficiency 

15 

Since the efficiency of the 1-inch steel rivet, 7^-inch pitch, was 
found in Art. 39 to be 15. 7 per cent, the efficiency, 15-inch pitch, 
may be found by proportion, which, in this case, is 2 to 1; hence 
the efficiency of the rivet in single shear, maximum pitch or 15-inch 
pitch, is 15.7-*-2 = 7.85 per cent. 

This may also be found as follows: 

.7854 x 45,000 x 100 
-= 7.85 efficiency 

60,000 x .5 x 15 

Adding the efficiency (7.85) to the efficiency of the net section of 
plate, 72-inch pitch, which, according to Art. 37, is 86.6 per cent, makes 
the combined efficiency of the net section of plate, 7J-inch pitch, and 
one rivet in single shear, 86.6 + 7.85 = 94.45 efficiency. As 94.45 effi¬ 
ciency exceeds the 93.3 efficiency of the net section of plate, 15-inch 
pitch, as given in Art. 43, the efficiency of the riveted joint (Fig. 15) 
is 93.3 per cent; the efficiency of the riveted joint (Fig. 14) is 86.6 
per cent. 




BOILER CONSTRUCTION. 


25 


PRACTICAL CONSIDERATIONS WHEN DESIGNING A RIVETED 

JOINT. 

45. With double-riveted lap joints an item that must be taken into 
consideration is to have the arrangement of the rivets such that pitch 
adjoining the calking edge is uniform. The rivets should be arranged, 
as shown in Fig. 16, and never as shown in Fig. 17. The difference 
in the respective arrangement of the rivets is that in Fig. 16 the greater 
number of rivets are in the outer two, while in Fig. 17 the reverse 
exists. With a single-riveted lap- joint it is customary to make 
the pitch a (Fig. 18) greater than the pitch b. This is done to per¬ 
mit the rivet to be readily inserted and driven, and also to permit 
the seam c to be readily calked. 

46. When a triple-riveted double-strapped butt joint, as shown in 
Fig. 19 is used the overall distance between the girth seams plays no 
little part in the maximum pitch. Theoretical calculations are many 
times badly upset, and all arisings from the fact that the overall dis- 
tance between the girth seams may be such that the pitch selected 
can not be used. In Fig. 19 is given a concrete case, and, as will 
be noted, the even number of rivets are in the outer row, per mini¬ 
mum pitch, thereby permitting the rivets in single shear to be so ar¬ 
ranged that the distance a is alike at both ends. 

Were one rivet added, or one rivet subtracted then the outer row 
of rivets, minimum pitch, would have an odd number of rivets, which 
would not permit arranging the rivet in single shear so that the distance 
a would be alike at both ends - and, needless to say, the joint would 
be anything but properly proportioned. The foregoing brings oift that 
the pitch may have to be altered, as the selected pitch may cause an 
odd number of rivets in the outer row, minimum pitch, and if this is 
the case another rivet must be added, decreaseing the pitch, or one 
rivet subtracted, increasing the pitch. 

47. The reason the distance a (Fig. 20), which is less than the 
distance b, is not computed is due to the fact that it derives assist¬ 
ance from the rivets in the girth seam. For the rupture to start 
through section a the marked rivets in the girth must be sheared, there¬ 
fore the force must not only break the net section of plate, but also 
shear the rivets in the girth seam. The assistance that the net sec¬ 
tion of plate derives from two rivets only, is generally sufficient to 
make its strength greater than the net section of plate b. 

FACTOR OF SAFETY. 

48. In the riveted joints figured out the supposition has been 
that the factor of safety was standard. However, the plate, depend¬ 
ing upon how the rivet holes are installed, may have a different factor 
of safety than the rivets. If the rivet holes are drilled, the plate should 
have a lower factor of safety than if the holes are punched. Whether 


26 


BOILER CONSTRUCTION. 


the rivet holes are punched or drilled has no marked bearing on the 
rivets, therefore, they may have a set or standard factor of safety. 
While it is true that rivets in punched holes show a greater shearing 
strength than rivets in drilled holes, this is due to the greater area. 

49. Let it be understood that the plate’s variable factor of safety 
is due to the manner of installing the rivet holes, also the condition 
of the holes. If unfair, that is, partly blind, a greater factor of safety 
is applied. Due to the foregoing it is possible for a riveted joint, es¬ 
pecially a lap joint, leaving the rivet efficiency less than the net sec¬ 
tion of plate to permit a greater working pressure than the net sec¬ 
tion of plate with a greater efficiency. 

For instance: If the efficiency of the net section of a triple-riveted 
lap joint is 75 per cent, factor of safety 5.5, and the rivet efficiency 
is 73 per cent, factor of safety 5, the working pressure found by the 
latter will be the greater of the two. In such cases the working pres¬ 
sure must be worked out, per respective efficiencies and factors of 
safety, and the least sum of the two methods is the allowable working 
pressure. The following will show the situation very clearly: 

50. Assuming the boiler to be 60-inch diameter, J-inch plate, 60,- 
000 tensile strength, and with triple-riveted lap joint, per respective 
efficiencies and factors of safety given in Art. 49, the allowable work¬ 
ing pressure would be the least of the following: 

Plate efficiency. 

60,000 x .75 x 100 

-= 136 pounds working pressure 

60 x 5.5 

Rivet efficiency. 

60,000 x.73x100 

-= 146 pounds working pressure 

60x5 

As will be noted the lower efficiency with the lower factor of 
safety shows a greater working pressure than the higher efficiency 
with the higher factor of safety. The working pressure allowable in 
this case is 136 pounds. Some, however, make a mistake and use the 
low efficiency and high factor of safety, and by so doing the work¬ 
ing pressure would figure out: 

60,000 x .73 x 100 

-= 132 pounds working pressure 

60 x 5.5 

The need of thoroughly understanding and employing the right 
factor of safety, when two different factors of safety are used, with 
the right efficiency, is apparent. 





BOILER CONSTRUCTION. 


27 


STAYING OF SURFACES 


ARRANGEMENT OF STAYS 

51. For the purpose of supporting various parts of a steam boiler 
and other structures, so that deformation of the plates will not take place 
when subject to a given load, stays and braces are used. Some parts of a 
steam boiler and other structures are of that shape that the pressure 
instead of causing distortion of shape, tends to maintain it. Under cir¬ 
cumstances of this character, bracing is not required. 

Braces may be classified into two kinds, direct and indirect. A direct 
stay or brace is one that is placed at 90° to the sheet it supports, and an 
indirect stay or brace is located at other than 90° to the surface it supports. 
All staybolts, radial bolts and crown bolts should always be at right 
angles to the firebox sheets, and should be so placed regardless of the 
angle to which the stays are attached to the outside wrapper sheet, though 
a boiler properly designed will provide that no indirect stay or brace will 
have an angle of over 20°. 

Every brace should be as direct as possible to the surface it supports; 
the greater the angle the less load the brace is allowed. The stress per 
square inch varies according to the material of which the brace is com¬ 
posed, and the manner in which the brace has been constructed. Stay 
bolts and welded iron braces are allowed by most authorities 6,000 pounds 
per square inch when subject to a direct pull, while steel braces without 
welds are allowed by different authorities from 7,000 to 9,000 pounds per 
square inch when subject to direct pull. 

The variation of the allowable stress per square inch is due to the 
factor of safety, which to some extent is regulated by the size of the brace. 
The factor of safety stay bolts and braces is greater than the factor of 
safety of the shell, and ranges from 6 to 10, according to the author¬ 
ities. The factor of safety of stay bolts and braces is made greater than 
other parts of the boiler as the braces are subject to corrosion, and are 
weakened by it more than any other part. The stay bolts, such as used 
in locomotive boilers and similar types of boilers, are not only subject to 
the load tending to separate the parts of the boiler which the staybolts 
supports, but are subject to a vibratory stress, which is caused by the 
expansion and contraction of the boiler-the vibratory stress breaks the 
staybolts. 

COMPUTING THE AREA. 

52. The area of a threaded stay (the staybolt the same diameter 
throughout its entire length) is computed from the root of the threads, 



28 


BOILER CONSTRUCTION. 


and not from the outside diameter of the stay bolt. If the threads are 
turned off along the body of the stay bolt, and slightly below the root of 
the threads, as they are in a great many instances, then the area is com¬ 
puted from the least diameter at any point throughout the length of the 
stay bolt. 

The area at the root of the threads depends as to the style of thread 
that has been cut on the stay bolt. A stay bolt having the Sharp V 
thread will have less diameter at the root than a stay bolt having the 
United States thread. The diameter of a l-inc:h stay bolt, Sharp V thread, 
is .85567 inch in diameter at the root of the threads, while a 1-inch stay 
bolt, United States thread, is .89175 inch in diameter at the root of the 
threads. The difference in diameter is slight, but sufficient to permit one 
stay bolt to have a greater working stress than the other, though both 
are 1-inch threaded stay bolts. 

53. The area and allowable working stress of a 1-inch threaded stay 
bolt, United States thread, is as follows: 

1 - .10825 = .89175 inch in diameter at root of the threads. 

.89175 x.89175 x.7854 = 62455 square inches. 

.62455x6000 (allowable stress in pounds per square inch) =3747 
pounds working stress. 

54. The area and allowable working stress of a 1-inch threaded stay 
bolt, Sharp V thread, is as follows: 

1 - .14433 = 85567 inch in diameter at root of the threads. 

.85567 X .85567 x .7854 = .57504 square inches. 

.5750x6000 = 6000 3450 pounds working stress. 

The difference in the allowable working stress of the foregoing in 
favor of the stay bolt, United States thread, is as follows: 

3747-3450 = 297 pounds. 


PROPORTION OF PARTS. 

55. Braces consisting of two or more parts, such as the body, palm, 
jaw and eye, should be so designed that the body is the weaker part of 
all parts. The designing is based on ratio and proportion. For instance: 
Assuming the diameter of the body of the brace Fig. 21, to be 1 inch, the 
area would be: li x 1£ x .7854 = .994 square inches-and if the brace was 
allowed 6000 pounds stress per square inch, then the actual load allowed 
on the brace would be: .994x6000 = 5964 pounds. 

When designing the palm of the brace it is necessary to consider 
that its minimum cross-sectional area must at least equal the minimum 
cross-sectional area of the body, and if the material is not homogenous 
the minimum cross-sectional area of the palm should exceed the minimum 


BOILER CONSTRUCTION. 


29 


cross-sectional area of the body. The force acting on the body in the 
direction is pulling with the grain, while the force acting on the palm 
is pulling across the grain. 



56. Material that is homogeneous can withstand the same stress 
across the grain as with the grain, or nearly so, the difference being so 
slight that it is not considered. Grain as herein used means the direction 
in which the plate or bar was rolled. The commercial steam boiler ma¬ 
terial is made as near homogeneous as practicable, and is spoken of as 
having no grain. The foregoing expression, however, merely means that 
the material is able to resist as great a force in one direction as in 
another direction. 

57. The body of the brace being li-inches in diameter, as mentioned 
in Art 55, and having a cross-sectional area of .994 inch, then the least 
cross-sectional area of the palm when made of homogeneous material must 
be at least .994 inch-say 1 inch. Assuming the thickness of fhe palm to 
be i inch, and the rivet hole 1 inch in diameter, the width of the palm 
must be sufficient so that there will be at least 1 in cross-sectional area. 
The width of the palm is found as follows: 

1-^| = 2 inches. 2x1 = 3 inches. 

Or the calculations may be written as follows: 

(1-T-|) +1 = 3 inches. 

58. If the width be first decided and the thickness of the brace is to 
be found, then subtract from the width of the palm the diameter of the 
rivet hole in the palm, after which divide the cross-sectional area re¬ 
quired by the above product. Assuming the measurements to be the 
same as in Art. 57, then the calculations can be written as follows: 

l-h(3 - 1) = I inch. 

59. Iron is not homogeneous and will part more readily across the 
grain than with the grain. If the brace, Fig. 21, is made of iron the 








30 


BOILER CONSTRUCTION. 


cross-sectional area of palm must be greater than the cross-sectional area 
of the body, and because the force acting on the body is pulling with the 
grain, while with the palm the force is pulling across the grain. To de¬ 
termine just how much more area is required in the palm (the force 
acting across the grain) than in the body of the brace (the force acting 
with the grain) requires only the use of ratio and proportion, and the 
ratio depends upon the grade of iron. No set ratio can be given, though 
the ratio will be near to 8 to 7. 

60. Assuming this ratio to have been established as correct in this 
instance, then the y 2 inch thickness of the palm as given in Art. 58, would 
not be sufficient; the correct thickness would be: 

£x8 

-= .57 inch. 

7 

Note —In practice i-inch material would be used. 

PITCH OF STAY BOLTS. 

61. The pitch of stay bolts-that is, the distance from center to 
center, depends upon the following: 

First: Size of stay bolt. Second: Thickness of plate the stay bolt 
supports. Third: The working pressure per square inch. 

The area to be supported by a stay bolt will be large or small accord¬ 
ing to the pitch. If the stay bolts are 4-inch centers, then one stay bolt 
supports 4 x 4 = 16 square inches. In order that the pitch may be the de« 
sired amount, the size of the stay bolt must be such as to be able to carry 
the load to which it will be subjected, and the plate must be of sufficient 
thickness that the pressure will not bulge it between the stay bolts. 

Assuming the pitch to be decided - say 4-inch pitch, which means one 
stay bolt supports an area of 16 inches - then the size of stay bolt needed 
for said pitch must be found. The load carried by the stay bolt depends 
upon the area it supports and the working pressure. If the stay bolt 
supports an rea of 16 inches and the working pressure is 150 pounds 
to the square inch, then the load the staybolt will have to withstand will 
be 16x150 = 2400 pounds. 

62. From the above it will be seen that the pitch, the area to be 
supported, and the load of the stay bolt is known, but the diameter of 
the later is yet to be found. Assuming the allowable working stress of 
the stay bolt to be 6000 pounds per square inch, then the medium area of 
the stay bolt would be: 

40 

-= .40 square inch 



BOILER CONSTRUCTION. 


31 


The diameter may then be found as follows: 

/ 40 

/-= . 71 inch 

V .7854 

63. These calculations show how to figure out the area required; 
also, the diameter of the stay bolt at the root of the thread. The outside 
diameter of the stay bolt depends upon the type of thread selected. As¬ 
suming the stay bolt to have United States thread, then add the constant, 
.10825, to the .71-inch, or .10825 + .71 = .81825 inch. However, .81825 inch, 
when changed to a fraction is a little greater than if inch, therefore, 
in this instance a I-inch stay bolt, which is about as small as should be 
used, permits (considering the stay bolts only) a working pressure greater 
than 150 pounds to the square inch on the boiler. The allowance pressure 
(stay bolts 4-inch centers) is found by multiplying the area of the |-inch 
stay bolt by 6000, and dividing the product by the area supported by the 
stay bolt, or: 

6000 x.4622 

-- 173 pounds 

16 

Note. —The decimal, .4622, as given in the foregoing is the area of a 
1-inch stay bolt, United States thread. 

RELATION OF BRACE TO PLATE. 

65. The arrangement of stay bolts and braces can not be spaced 
without regards as to the thickness of the plates which the stay bolts or 
braces are to support-in fact the whole purpose of staybolts and braces 
is to support the plate in such a manner the deformation will not take 
place. While the stay bolt or brace may be of sufficient size to carry the 
desired load, the allowable pressure on the plate is regulated more or less 
by the method of attaching the stay bolt or brace to it, and the manner 
of attaching the stay bolt or brace may be such the pressure allowed will 
be considered below the pressure desired. This might result in a heavier 
plate being used; or a change in the method of fastening the stay bolts or 
braces; or a change in the pitch, which may cause smaller stay bolts or 
braces to be used. 

66. A firebox sheet is of light material to permit rapid generation of 
steam; the heavier the plate the more difficult to generate steam. The 
method of fastening stay bolts and braces depends upon their location in 
regards to the flames and hot gases. The stay bolts, which support the 
locomotive firebox side sheets, are secured to the sheets by being 
screwed into the plates; a portion of the bolt projecting beyond the plate, 
and then riveted over. 

67. The allowable working pressure for the plate depends upon its 
thickness; the pitch of the stay bolts; and the method of fastening the 









32 


BOILER CONSTRUCTION. 


stay bolts or braces. The latter is regulated by a constant, and varies 
acording to circumstances. The United States Steamboat Inspection Ser¬ 
vice authorizes with screwed stays, riveted over, the constant 112 for 
plates &-inch and lighter, and the constant 120 for plates heavier than 
T Vinch. 


The allowable working pressure for the plate can be determined by 
the following formula: 


A x D 
-= B 

C 


Where: 


A = Constant. 

B = Pressure in pounds per square inch. 

C = Maximum pitch of stay bolts in inches. 

D = Thickness of plate in sixteenths of an inch. 

68. Assuming the thickness of the side sheets to be tfe-inch (which 
is about as thin as used with a high pressure boiler), the pressure allowed 
on the plate, stay bolts 4-inch centers, is found by subtracting values, or: 
7 x 25 = 175 pounds working pressure. 

7 

m X 25 

-= 175 pounds working pressure 

69. Comparison of the foregoing with the product obtained in Art. 
64, shows that the allowable working pressure on the plate and the allow¬ 
able working pressure on the stay bolt is about the same, per following 
conditions: Constant 112; 1-inch stay bolt, United States thread; t^-inch 
steel plate, and stay bolts 4-inch centers. 

70. When the pressure and the thickness of the plate are known 
(which are usually the two things first determined), then the pitch of the 
stay bolts can be found by the following formula; the letters in the 
formula represent the same values as given in Art. 67: 


/A x D 

i/ — l~ = c 

Substituting values: 

/16 

/ 

/ -= 4 inch centers 


m 








BOILER CONSTRUCTION. 


33 


71. It is not always practicable to make the pitch of the stay- 
bolts uniform-that is to say, the horizontal distance between the stay- 
bolts can not always be made the same as the vertical distance be¬ 
tween the staybolts. However, the respective distances should be as 
near uniform as practicable, and the area the staybolt is to support is 
figured by some as the square of the maximum pitch. Thus, if the 
pitch of the staybolts is 4 x 31 inches, the area is computed as 4x4 =16 
square inches. 

72. Many locomotive boilers are designed so that the outer row 
of staybolts are of greater diameter than the staybolts of the inner 
rows. The principal reason of the foregoing is that the shape of the 
boiler is such that the staybolts at certain places must be placed excep¬ 
tionally far apart. For instance: The pitch of staybolts in the illus¬ 
tration, Fig. 22, may be uniform throughout except the distance a be¬ 
tween the line of rivet holes of the mud ring and the first row of stay- 
bolts above the mud ring. 



73. The distance a is equal to the distance 6, plus the distance c. 
Assuming the pitches b and d to be 4 inches, and the pitch a to be 
5 inches, then the actual area to be supported by the stayblot next 
to the mud-ring would be 4 x 5 = 20 square inches. However, some au¬ 
thorities figure the area as 5x5 = 25 square inches, while others fig¬ 
ure the area as follows: 

42 x 52 41 


2 


2 


= 20| square inches. 



















34 


BOILER CONSTRUCTION. 


The foregoing merely gives the different areas computed by dif¬ 
ferent methods. As there is no general understanding among mechan¬ 
ical men as to which method should be used in practice, one designer 
uses one method and another designer another method, and though the 
boilers designed by the respective parties may be alike in construction, 
the working pressure allowed would be greater in one case than in 
the other. 

74. In Arts. 64 and 68 it was shown that if the staybolts were 4-inch 
centers, plates tfc-inch thick, the allowable working pressure for the 
staybolts would be 173 pounds per square inch, and for the plate 175 
pounds per square inch. However, if the boiler was of the locomotive 
type, requiring the staybolts to be arranged as shown in Fig. 22, mak¬ 
ing the pitch a 5 inches, while the pitches h and d were each 4 inches, 
the allowable working pressure, ^-inch plate, would be a considerable 
less than stated in Arts. 64 and 68. 

75. Just how much less the pressure would be depends upon how 
the area is computed. If the maximum pitch times the minimum pitch 
method is employed, or 4x5 = 20 square inches, then the allowable work¬ 
ing pressure for the plate will be: 


28 5 

-= 140 pounds 

W 

4 


76. If the method of squaring the maximum pitch is used, or 
5x5 = 25 square inches, then the allowable working pressure will be: 


112 

-= 112 pounds 

n 


77. In Art. 64 the allowable pressure for a 1-inch staybolt, 4-inch 
pitch, tk-inch plate, was found to be 173 pounds. Since the pitch a, 
Fig. 22, is 5 inches, the area the staybolt must support is increased. 
Assuming the area to be 20 square inches, as given in Art. 75, then 
the allowable working pressure for a 1-inch staybolt will be: 

300 

x .4622 

-- = 138.66 pounds 

w 

78. Assuming the area to be 25 square inches as given in Art. 76, 
the allowable working pressure for a 1-inch staybolt will be: 

240 

W0 x. 4622 


= 110 pounds 






BOILER CONSTRUCTION. 


35 


79. The foregoing well illustrates the need of the mechanical men 
getting together and adopting a standard. The constant, 112, as given 
in Art. 64 and 69, is not an accepted standard. The calculations, how¬ 
ever, show that the plate in each instance is allowed the greatest work¬ 
ing pressure. 

80. Assuming the area to he supported by the staybolt to be 20 
square inches, 1-inch staybolt, ^-inch plate, which (see Art. 75) 
permits, for the plate an allowable working pressure of 140 pounds to 
the square inch, and if this is the working pressure desired, then the 
staybolts in the row adjacent to the mud ring would have to be greater 
than 1-inch diameter, for, according to Art. 77, the allowable work¬ 
ing pressure is 138 pounds to the square inch. A larger staybolt—say 
tfr-inch in diameter - would suffice in this case, and would permit 
an allowable working pressure of over 140 pounds to the square inch. 
While the boiler may be constructed so that many parts could be al¬ 
lowed a greater working pressure per square inch than another part, 
the working pressure allowed must be computed from the weakest point 
-no chain is stronger than its weakest link. 



AREA SUPPORTED BY A RIVET. 

81. When boilers are constructed with T-irons, or angle irons, 
riveted to the back head, front flue sheet, etc., for the purpose of sup- 































36 


BOILER CONSTRUCTION. 


porting the sheet, and providing means for attaching the braces, the 
size and pitch of the rivets used for attaching the T-iron, angle irons, 
etc., to the plates have a bearing on the allowable working pressure. 
The back head and the front flue sheet are usually of very heavy ma¬ 
terial, and generally the only consideration is the size and pitch of 
the rivets which attach the T-irons, or angle irons to the plates. The 
pitch of the rivets is usually irregular - that is to say, vary, which is 
well illustrated in Fig. 23. 

82. It is the practice with most of the designers to figure that 
the rivet is to support an area equal to the square of the maximum 
pitch. Thus, if the pitch a, Fig. 23, is 5 inches, and the pitch b, 4 
inches, the area is assumed to be: 

5x5 = 25 square inches. 

The area is then multiplied by the working pressure-and assum¬ 
ing this to be 140 pounds-the same as found in Art. 75-then the as¬ 
sumed load carried by the rivet would be: 

25x140 = 3,500 pounds. 

83. The rivets, in this instance, would be subject to practically 
a direct pull-would be in tension. Allowing a working stress of 6,000 
pounds per square inch, which is the same as allowed for a direct pull 
on the threaded staybolt, the area of the rivet required would be: 

3500-T-6000 = .5833 square inch. 

The area would be: 



/ .5833 
.7854 


.87 - say g-inch. 


84. The actual load allowed on a g-inch rivet figuring on 6000 
pounds allowable stress per square inch, would be: 

.6013 x 6000 = 3607 pounds. 

As the load on a g-inch rivet, 5-inch pitch, 140 pounds working 
pressure, was found in Art. 82 to be 3500 pounds, and as the rivet (See 
Art. 84) is allowed a stress of 3607 pounds, the g-inch rivet in this 
instance is sufficient for the purpose. 


DISTRIBUTION OF BRACES. 

85. Many backheads and front flue sheets of locomotive boilers, 
and the two heads of tubular boilers, etc., are stayed by direct and in¬ 
direct stays attached to the head as shown in Fig 24. When arrang- 



BOILER CONSTRUCTION. 


37 


ing the stays, staying an irregular surface, it is practically impossible 
to so distribute the stays that each one will carry the same load. 

A portion of the head of a tubular boiler is shown in Fig. 24, and 
the letters a and b represent the pitch of the braces in the respective 
rows. The pitch a, which is the distance from center to center of the 
braces of the outer row, is greater than the pitch b, and for the rea¬ 
son that the radii in the former is greater than the latter. It may 
be possible in some cases to put less braces in the inner row than in 
the outer row, but the pitch of the braces of the respective rows will 
not be the same. 

Under these conditions the area supported by the brace must be 
computed from the maximum pitch, which in Fig. 24 is the distance a. 
Braces constructed with a crowfoot are usually attached to the plates 
with two rivets. The pitch between the two rivets is usually uniform 
regardless of the pitch of the braces. Thus, the distance c, Fig. 24, 
varies with the pitch of the braces, and the distance d, which is the 
pitch of the rivets in the crowfoot of the braces, is the same in all 
the braces, both outer and inner rows. 

The area supported by the brace is: 

Where: 


a = maximum pitch of braces in inches. 
e = distance between rows or braces in inches. 

axe- area. 

Assuming the pitch a to be 9 inches and the distance e to be 6 
inches, then the actual area supported by the brace would be: 

6x9 = 54 square inches. 

86. As stated in Art. 73 authorities differ as to the manner of fig¬ 
uring the area, and some, perhaps, would square the maximum pitch, 
or 9 x 9 = 81 square inches. However, when a boiler head is braced 
as shown in Fig. 24, the area supported by the brace should be figured 
as the maximum pitch times the minimum pitch. The load carried by 
the inner row of braces will not be as great as the load carried by the 
outer row of braces- and while some designers under such conditions 
would use a smaller size brace for the inner row than the outer row, 
the general practice is to use the same size brace in both the inner 
and outer rows. 

87. The area supported by the rivets, Fig. 24, must be figured as 
c x e instead of d x e. In Art. 85 it was assumed that the distance a 
was 9 inches and the distance e was 6 inches, making an area of 54 
square inches to be supported by the brace. Since each brace has two 
rivets the first conclusion would be that each rivet carried one-half of 
the load, and, therefore, supports 27 square inches. 


38 


BOILER CONSTRUCTION. 


However, referring to Art. 82 it will be seen that the practice with 
most designers is to figure the area supported by the rivet as< equal 
to the square of the maximum pitch, and this in this case would be: 

6x6 = 36 square inches. 



Some designers, however, when the braces are arranged as shown 
in Fig. 24, make an exception to the general rule, and figure the area 
by multiplying the maximum pitch by the minimum pitch. Then as¬ 
suming the distance d, Fig. 24, to be 4 inches, which makes the dis¬ 
tance c, 5 inches, the area supported by the rivet would be: 

5 x 6 = 30 square inches. 

SUPPORTING THE SHEETS. 

88. As stated in Art. 67 the constant used in connection with tne 
method of fastening the braces varied according to circumstances and 
authorities. Frequently a few large braces instead of a number of 
small braces are used, but in such cases the plate is reinforced by 
another plate. For instance: The backhead of a locomotive type boiler 
may be f-inch, i-inch staybolts at 4-inch. centers, and the longitudinal 
braces which support the upper part of the head may be spaced 6 by 
9 inches. Therefore, the plate between the staybolts, using the con¬ 
stant 112 as given in Art. 67, would permit a working pressure as fol¬ 
lows: 






BOILER CONSTRUCTION. 


39 


7 

m X 36 

-= 252 pounds. 

IV 

The area to be supported by a 1-incb staybolt would be 16 square 
inches, and the allowable working pressure as far as the plate is con¬ 
cerned is 252 pounds. In Art. 68 it was shown that with the same 
size and pitch of stayholts as given in the above example, the allow¬ 
able working pressure for the plate would be 173 pounds, and in Art. 
64 it was shown that a 1-inch staybolt, 4-inch centers, regardless of 173 
the thickness of the plate, would be allowed a working pressure of 173 
pounds. 

89. To space the longitudinal braces, 6 by 9 inches, without rein¬ 
forcing the head at that portion would reduce the allowable working 
pressure, or as follows: 


2 

112 x W 

-= 74 pounds. 

H 

3 

The foregoing calculations show that the head where the longitu¬ 
dinal braces are attached must be heavier than where the staybolts are 
attached. This is accomplished by a reinforcing plate. Assuming a 
|-inch reinforcing plate to be added to the |-inch head, then the thick¬ 
ness of both would be i-inch. However, when a reinforcing plate is add¬ 
ed to the head in the manner described, most designers when figuring 
out the allowable working pressure assume the thickness to be about 
75 per cent of the actual thickness. Two plates secured together by rivet¬ 
ing are not able to resists the same as one plate of the same thickness 
as the two plates. Under these conditions two 1-inch plates would be 
considered in the calculations as the same as a f g -inch plate. Then the 
allowable working pressure would be: 

20 9 

m x n 

-= 180 pounds. 

H 

6 

Note —Since the plate is over ^-inch in thickness the constant 
120 (See Art. 67) is used. 

90. A study of the underlying principles of boiler designing will 
show the object of reinforcing certain parts of a boiler. In the fore¬ 
going case the addition of the g-inch reinforcing plate increased the 
allowable working pressure (See Art. 89) from 74 pounds to 180 pounds, 
and caused to be used a constant with greater numerals. The furnace 
sheet is generally considerably lighter than the outside sheets. If 





40 


BOILER CONSTRUCTION. 


the outside sheet is very heavy - say §-inch - and the furnace sheet 
light-say ^-inch-the allowable working pressure as far as the out¬ 
side plate is concerned will be many pounds greater than the furnace 
sheet. The allowable working pressure of the boiler, however, is the 
lowest working pressure allowed for any part. Each part must be 
computed in order to ascertain just what part is the weakest of all 
the parts. The construction of the boiler may be such that one part 
is twice or three times as strong as another part. The factor of safety 
for all parts is not alike, nor is the constant, etc. All the foregoing 
must be considered when designing a boiler. 


BRACING CURVED SURFACES 

i 

STAYING A FURNACE. 

91. The sheet a, Fig. 25, must be stayed the same as a flat sheet. 
Though the sheet is curved, the pressure tends to crush in or collapse 
it; the force in this instance acting the opposite to its action on the 
boiler shell. In Fig. 2, the forces tending to rupture the shell by internal 
pressure are shown by arrows, but the force acting on the crown sheet 
a, Fig. 25, is an external force; hence, the radial stays b, c, d, and e, are 
for the sole purpose of supporting the crown sheet a, and not the roof 
sheet a\ The side staybolts below the line A A are for the purpose of 
staying both the furnace side sheet / and the outside wrapper sheet g. 

As the crown sheet of most of the locomotive boilers is higher at the 
front (firebox flue sheet end) than at the back (door sheet end) the 
radial and crown bolts supporting the crown sheet should be at right 
angles to it irrespective of how they strike the roof sheet. If the roof 
sheet is not heavy and the stays strike it at such an angle that a suf¬ 
ficient number of continuous threads, equal to the amount in the crown 
sheet, cannot be secured, then a reinforcing line h, should be placed 
inside the roof sheet as shown in Fig. 25. 

92. Inspection of Fig. 25, reveals that the stays radiate from the 
same points as used as centers with the respective arcs for developing 
the furnace. Thus: the stays b, c, and d, radiate from the point i ; the 
stay e, from the point ;; and some of the other stays radiate from the 
points k, and l. The distance between the rows of stays of the furnace 
sheet /, is uniform, but not so with the outside sheet g ; also, the distance 
between the rows of stays of the roof sheet will vary considerable. 

ALLOWABLE LOAD ON AN INDIRECT BRACE. 

93. Every brace should be as direct as possible - that is to say, 
should be as near at a right angle as possible to the surface it supports. 



BOILER CONSTRUCTION. 


41 


It is impossible, however, to locate all braces at right angles to the 
surfaces they support; hence, the indirect brace, though no indirect 
brAce should be placed over 20° to the surface it supports. 



Like many other parts of a boiler, the allowable load on an indirect 
brace is figured differently by authorities. An indirect brace and a 
direct brace of the same size and construction are allowed different 
working stresses-the direct brace is allowed a greater stress than the 
indirect brace. 

94. For instance: The body of the brace a, Fig. 26, is 1£ inch in 
diameter, which makes an area of .994 square inches. The brace is in¬ 
clined at the maximum-20°-and to determine the allowable working 
stress, figuring 6,000 stress per square inch, direct pull, the following 
formula, which is one of the many, is used. 

















42 


BOILER CONSTRUCTION. 


Where: 

a = Length of brace in inches in horizontal plane, 
b = Length of brace in inches on the incline. 
c= Allowable working stress per square inch, direct pull, in 
pounds. 

d = Allowable working stress per square inch, direct pull, in 
pounds. 

c = Area of brace in square inches. 

axcxe 

-= d 

b 

95. Assuming the distance a, Fig. 26, to be 47 inches, and the dis¬ 
tance b, to be 48 inches, then substituting values the allowable working 



stress for the brace, 1§ inch in diameter, allowable working stress of 
6000 pounds per square inch, direct pull, is: 

125 

47 x x .994 

-— 5839.75 pounds 

COLLAPSING PRESSURE OF TUBES. 

96. The tubes of all boilers except water-tube boilers, are subjected 
to external pressure. The pressure is as great at one point as at another 
point, and accordingly it would appear reasonable to presume that the 
pressure on one side would offset the pressure on the other side. How- 
























BOILER CONSTRUCTION. 


43 


ever, the walls of the tube are not strictly uniform, nor is a tube a true 
cylinder. 

The internal pressure tends at all times to round out the vessel 
to a true cylinder, while the external pressure is just the reverse. As 
soon as the tube assumes any shape other than that of a perfectly true 
cylinder, it is easy prey to the pressure and the latter causes the col¬ 
lapsing of the tube. For this reason it is essential that tubes and fur¬ 
naces that are subjected to external pressure be made as perfectly true 
in diameter as possible. The formation of scale on a cylinder subjected 
to external pressure, will cause a greater pressure at one point than at 
another point, and this coupled together with the fact that the working 
of the boiler causes shocks, leads up to the collapsing of many flues; 
also, round furnaces such as used in marine and other types of boilers. 

97. Professor Reid T. Stewart, of Allegheny, Pennsylvania, con¬ 
ducted experiments to ascertain the collapsing pressure of flues. Prior 
to his experiments it was the general practice throughout the country to 
take into consideration the length of the tube or furnace from end to end, 
ring to ring, or joint to joint. The experiments showed that this was a 
false theory with regards to flues, for a long flue may collapse at one 
point and the balance of the flue be perfectly true. The support from 
the rigid end was so insignificant that it was not worthy of being taken 
into consideration. 

Professor Stewart, after making many tests advanced formulas A 
and B. 

Formula A: 


P = 1000 



1-1600 


(T)2 \ 

(DP ) 


Formula B: 


Where: 


T 

P= 86,670-= 1,386. 

D 


P = Collapsing pressure in pounds per square inch. 

D = Outside diameter of tube in inches. 

T = Thickness of wall in inches. 

Note. —Formula A is for values less than 581 pounds, or for values 
of T-t-D less than .023. Formula B is for values greater than these. 




44 


BOILER CONSTRUCTION. 


TABLE II. 

The United States Steam-boat Inspection Service allows 225 pounds 
pressure per square inch on all lap-welded flues up to 6 inches in diam¬ 
eter, if the material conforms to Table II. 

Outside Outside 

Diameter Thickness Diameter Thickness 


1 inch 

.072 inch 

3 inch 

.109 

inch 

11 inch 

.072 inch 

31 inch 

.120 

inch 

lh inch 

.083 inch 

3i inch 

.120 

inch 

11 inch 

.095 inch 

31 inch 

.120 

inch 

2 inch 

.095 inch 

4 inch 

.134 

inch 

21 inch 

.095 inch 

4h inch 

.134 

inch 

21 inch 

.109 inch 

5 inch 

.148 

inch 

21 inch 

.109 inch 

6 inch 

.165 

inch 


98. Marine boilers are generally constructed with a round furnace, 
which is subjected to an external pressure. Unlike small flues the length 
of the furnace must be considered when computing the allowable work¬ 
ing pressure. The greater the diameter of the furnace, the heavier must 
be its walls, though to generate steam the thickness of the walls must 



be kept within the limits. For this reason furnaces are made corrugated, 
the corrugations act as a rib or a supporter to the furnace, thereby 
permitting the use of a furnace of a large diameter and thin walls than 
would be permissable if the furnace was a plain round cylinder. Fur¬ 
naces are, however, constructed as a plain cylinder, but are re-inforced 
by ribs, etc., or the furnace is constructed in sections, called RINGS, as 
shown in Fig. 27. 

The length of the furnace is taken into consideration as the furnace 
receives more or less support from the rigid ends. While the support 



























BOILER CONSTRUCTION. 


45 


from the rigid ends is not distributed uniformly over the furnace, it aids 
the entire furnace between the rigid ends, the center the least, unless the 
distance between the rigid ends is exceptionally great, and in such 
cases the center receives the support. The allowable over-all distance of 
the furnace between rigid ends depends upon the diameter of the 
furnace, the thickness of the walls, the style of furnace, etc. 

99. A formula advocated when the furnace is constructed as shown 
in Fig. 27, sections not less than 18 inches in length and walls not less 
than ^-inch thick, and flanged to a depth of not less than three times 
the diameter of the rivet hole plus the radius at furnace wall (inside 
diameter of the furnace), the thickness of the flanges to be as near the 
thickness of the body of the plate as practicable, is as follows: 

51.5 

P =-[18.75x8 - (48x103) ] = 


D 


Where: 


P = Working pressure in pounds per square inch. 

D = Outside diameter of the furnace in inches. 

L = Length of the furnace in inches. 

T = Thickness of the plate in sixteenths of an inch. 

Example: 

Assuming a furnace to be 44 inches in diameter, 48 inches in length* 
and the walls i inch thick, then substituting values in the formula, the 
allowable working pressure will be: 


51.5 


P — - 


[18.75 xT- (Lx 1.03) ] 


44 


1.17 [150-49.4] =117.7 pounds. 

In calculating the allowable working pressure of a corrugated fur¬ 
nace the first thing to do is to look up the rules and regulations of the 
authorities under whose supervision the boiler is operated. The follow¬ 
ing formula for corrugated furnace is advanced by the Board of Super¬ 
vising Inspectors, United States Steam-boat Inspection Service: 


CxT 


P = - 


D 


Where: 


P = Pressure per square inch pounds. 

T = Thickness in inches (wall not to be less than ^-inch). 

D = Mean diameter in inches. 

C = Constant. 

Note. —The mean diameter means mid-way between the inside and 
outside diameters. Some authorities make D in the formula to represent 





46 


BOILER CONSTRUCTION. 


the outside diameter, while others make D to represent the inside diam¬ 
eter. The constant under such conditions naturally varies, and since 
there are several forms of corrugated furnaces, the type of furnace, 
together with the rules and regulations of the authorities under whose 
supervision the boiler will be operated, must always be considered. 

RE INFORCING RINGS. 

100. Tubular, marine and many other types of boilers are con¬ 
structed with a manhole for the purpose of permitting entrance to the 
boiler. A manhole should never be round, and for the reason that the 
manhole cover cannot enter the boiler and be placed in place. A man¬ 
hole is usually made about 15 inches by 11 inches, which is large enough 
to permit a man to enter the boiler. As the stress on the longitudinal 
plane is twice as great as the stress on the transverse plane, the man¬ 
hole should be placed so that the minor diameter of the manhole is in 
the longitudinal plane. 



To compensate the boiler shell for the strength lost by cutting the 
hole, a re-inforcing liner in placed around the hole as shown in Fig. 28. 
Assuming that the manhole is 11 inches by 15 inches, minor diameter 
placed in the longitudinal seam, or lengthwise of the boiler, the section 
of metal to replace would be 11 inches times the thickness. Assuming 
the shell to be ^-inch, then the area removed will be 11 x .4375 = 4.8125 
square inches. The re-inforcing liner may be of any thickness, but not 



























BOILER CONSTRUCTION. 


47 


less than the shell, and in this instance assuming the liner to be ^-inch 
thick, its width will then be: 

4.8125 

-=8.59 inches 

.562 

One-half of this will be on each side of the manhole, or 8.59-T-2 = 8.30 
inches. This, however, is the net section of plate, and as the liner must 
be riveted in place, the width of the liner must be increased by an amount 
equal to the diameter of the rivet used. Should 1 inch rivets be used, 
then the width of the liner on each side of the manhole would be: 
4.30-r-l = 5.30 inches. 

The liner, nowever, must be of the same material as the boiler 
shell, or at least have the same strength. Cast iron re-inforcing rings 
have gradually fallen into disuse, and because of the lack of homo¬ 
geneity in cast iron; also, the low tensile strength, and blow holes 
which are frequently found in iron castings. 

In computing the number of rivets to be placed in the re-inforcing 
liner, the usual practice is to make their total shearing strength not less 
than the total tensile strength of the re-inforcing liner. Previous cal¬ 
culations gave for the cross-sectional area, longtitudinal plane, 4.8125 
square inches, and assuming the tensile strength of the plate to be 
60,000 pounds per square inch, the total will be: 4.8125x 60,000 = 288,750 
pounds. 

With 1 inch rivet holes (area .7854), and assuming the shearing 
strength of the rivet to be 42,000 pounds per square inch, then one 1 inch 
rivet would have a shearing strength of 42,000 x .7854 = 32,986.8 pounds. 
The total rivets required would be: 

288,750 

--= 8f rivets 

32,986.6 

Note: In this instance the calculations show that 8 rivets, 1 inch 
diameter, are nearly sufficient. If 9 rivets, 1 inch in diameter, wer used, 
then the shearing strength of the rivets would exceed that of the net 
section of plate removed by considerable, but notwithstanding this, 
the number of rivets to be used in this instance would be 10, thus per¬ 
mitting 5 rivets to be placed on each side of the manhole as shown in 
Fig. 28. Since 10 rivets are to be used and their shearing strength ex¬ 
ceeds that of the net section of plate by many pounds, it will be seen 
that a rivet smaller than 1 inch can be used. The area of a 1^-inch 
rivet is .59 square inch, and using the same shearing strength as with the 
1 inch rivet, the shearing strength of a ^-inch rivet is: 42,000 x.69- 
28,980 pounds. Then, 288,750-^-28,980 = 10 rivets, thus, in this instance, 
10 rivets, £§-inch in diameter will suffice. 




48 


BOILER CONSTRUCTION. 


BOILER CAPACITY. 

HEAT AND STEAM. 

THE THEORY OF STEAM MAKING. 

101. All scientists agree that heat is a form of energy. It is well 
known that in order to change ice to water heat is required. By adding 
sufficient heat the water can be changed into steam. All matter is com¬ 
posed of molecules which are in a state of unrest, moving or vibrating 
back and forth with more or less velocity. If the motion is slow, the 
body feels cold; if the motion is rapid, the body feels warm, and it is 
this movement of the molecules that is regarded to cause the foregoing. 

Steam is merely water changed into a gaseous state by the appli¬ 
cation of heat. To generate steam it is first necessary to produce the 
heat and second to transfer the heat to the water. Water in an open 
vessel can be heated to 212° F., and no hotter. When it reaches that 
temperature the water boils and is converted in water vapor or steam. 

Heat cannot be measured directly in pounds, and, because it is not 
a substance, but it can be measured by the effects it produces. To 
raise the temperature of one pound of water one degree requires heat. 
The UNIT QUANTITY OF HEAT required to raise the temperature 
of a pound of water one degree is called a BRITISH THERMAL UNIT; 
generally abbreviated to B. T. U. 

While for ordinary purposes it is assumed that it takes one B. T. U. 
to produce a change of one degree in the temperature of one pound of 
water, it actually does, however, require slightly more than one B. T. U. 
to change the temperature above 63°—the difference increasing the 
farther the temperature is from 63°. Below the temperature of 62° it 
requires slightly less than one B. T. U. to produce a change of one 
degree in the temperature. From this it will be seen that it requires 
more heat to raise the temperature of one pound of water from 80° to 90° 
than it would to raise it from 88° to 89°; also, it will take less heat to raise 
the temperature of one pound of water from 41° to 42°. For all tem¬ 
peratures likely to be met in practice it is proper to assume that it 
requires one B. T. U. to raise the temperature of one pound of water 
one degree. 

When a given quantity of water at a given temperature has been 
changed into steam at a given pressure, a certain definite work has 
been done, and a certain amount of energy expended. As heat can be 
changed into work and work into heat, there must be a mechanical 
equivalent to one B. T. U. This was found to be 778 foot-pounds. 

Mix hot water with cold water and the result will be that the tem¬ 
perature of the cold water increases while the temperature of the hot 
water decreases. The exact temperature will depend upon the respective 


BOILER CONSTRUCTION. 


49 


quantities and temperatures of the hot and cold water. In an open 
vessel the temperature cannot be made greater than 212°, and for the 
reason that the steam can escape. In a closed vessel, the steam cannot 
escape, therefore, the temperature of the steam will increase according to 
the amount of heat applied. 

Water in an open vessel is exposed to the atmosphere pressure 
of 14.7 pounds per square inch (sea level - in practice generally assumed 
to be 15 pounds to the square inch) and boils at 212°. In a closed 
vessel the boiling point depends upon the steam pressure per square 
inch. As the pressure increases the boiling point raises accordingly. 
"Water, steam pressure six pounds per square inch, will boil at 170°, 
while water, steam pressure 32 pounds per square inch, will boil at 254°. 

Steam is spoken of as dry, wet, saturated and super-heated. If the 
steam is in contact with water, which is the condition of the steam in 
a boiler, then the steam is SATURATED STEAM. ’ When a boiler is 
steaming rapidly, the ebullition is liable to cause water in the form of a 
spray to mingle with the steam, and steam used with the water mingled 
is known as WET STEAM. When the steam is separated from the 
water, it may like air and other gases be heated higher than the boiling 
point corresponding to its pressure, and the steam is then known as 
SUPER-HEATED STEAM. 

, It is not possible to cool saturated steam except by lowering its 
temperature, nor can steam in contact with water be heated above a 
temperature normal to its pressure. Priming or wet steam is due to 
impure water, or two much water, or to improper proportions in the boiler. 
Steam which contains no water and is transparent, is DRY STEAM. 

HEATING SURFACE. 

102. The heating surface of a boiler is that part in contact with the 
flames and hot gases. With a locomotive type boiler, the heating surface 
includes the furnace, the flues, and the front flue sheet, though some 
do not consider the latter; with a tubular boiler, the heating surface 
includes that portion of the boiler shell and the two flue sheets exposed 
to the flames and hot gases; also, the flues. 

When figuring the heating surface of the flue sheet, be sure to 
deduct the area of the flue holes. The heating surface of a tubular boiler 
is generally calculated on the basis that two-thirds of the boiler shell is 
exposed to the flames and hot gases. The amount of the boiler shell so 
exposed will depend upon the location of the brackets, or if hangers are 
used, which are used extensively at present, upon the location of the 
hangers and the arrangement of the brick work below them. 

The heating surface of a flue is ascertained by multiplying its inside 
diameter by the constant 3.1416, times its length in inches. Thus the 


50 


BOILER CONSTRUCTION. 


heating surface of a 3-inch flue, i-inch thick, and 14 feet long would 
be: 3 - (2 x |) = 21-inch. 

2f x 3.1416 x 168 = 1451.52 square inches. 

In connection with the heating surface must be considered the grate 
area. The rate of consumption of fuel per square foot of grate surface 
depends upon the draft. With one type of boiler the rate of fuel con¬ 
sumption will be 15 pounds of coal per square foot of grate surface 
per hour, while with the locomotive boiler, where the proportion of 
grate surface to the heating surface is abnormally low, the rate of fuel 
consumption has been as high as 150 pounds of coal per square foot of 
grate area. The ordinary ratio of heatnig surface to grate area in dif¬ 
ferent types of boilers is as follows: 


Locomotive. 

.50 

and 

100 

to 

Water tube. 

.35 

and 

40 

to 

Tubular. 

.25 

and 

35 

to 

Vertical. 

.25 

and 

30 

to 

Flue type . 

.20 

and 

25 

to 


HORSEPOWER OF BOILERS. 

103. Literally speaking, there is no such thing as horespower to a 
steam boiler; it is a measure applicable only to dynamic effect. The term, 
however, has come into general use, and is applied to a boiler as boilers 
are necessary in order to drive steam engines, etc. Considering that 
the ordinary throttling and slide-valve engine uses 35 to 45 pounds of 
steam per horsepower per hour; the simple Corliss engine uses 24 to 30 
pounds of steam per horsepower per hour; the Corliss condensing uses 
19 to 22 pounds of steam per horsepower per hour; and the compound 
Corliss condensing 14 to 17 pounds of steam per horsepower per hour, 
it will be seen that the so-called standard boiler horsepower, which 
consists of the evaporation of 30 pounds of water per hour from a feed- 
water temperature of 100 degrees F. into steam at 70 pounds gauge 
pressure, may he a high or a low value per unit of work, depending upon 
the type of engine to which it is applied. 

Standard boiler horsepower rules based on evaporation and absorp¬ 
tion cannot always be conveniently used. At 125 steam pressure per 

Note.—The judges in charge of the boiler trials at the Centennial 
Exposition ascertained that a good engine of the then prevailing types 
required about 30 pounds of steam per hour per horsepower developed. 
They recommended that an expansion of 30 pounds of water per hour 
from an initial feed-water temperature of 100 degrees F. to steam at 70 
pounds gauge pressure be considered as one boiler horsepower. 

To permit comparison of results of the boiler trials, the usual prac¬ 
tice is to reduce them all to a basis of equivalent evaporation from and 
at 212 degrees. Then one boiler horsepower as above defined is equivalent 
to an evaporation from and at 212 degrees of practically 34.5 pounds. 







BOILER CONSTRUCTION. 


51 


square inch, the evaporation would be at a temperature of 344 degrees F. 
If the pressure were 200 pounds per square inch, the corresponding 
temperature of evaporation would be 381 degrees F. As some engineers 
rate the horsepower to a given number of square feet of heating surface, 
the following will serve for rough usage: 


Locomotive.12 and 16 to 1 (natural draft) 

Water tube.10 and 12 to 1 

Tubular.12 and 14 to 1 

Vertical.14 and 16 to 1 

Flue type. 8 and-12 to 1 


SAFETY VALVE. 

104. The safety valve should always be attached directly to the boil¬ 
er; it should be placed on the main steam pipe, or be so situated that it 
can be cut off at any time. The size of the safety valve should be suffi¬ 
cient to permit the free escape of the excess steam, discharging it so rap¬ 
idly that very little or if any increase in the pressure of the steam can 
take place regardless of how fast the steam is generated. The safety valve 
should have a large area for the escape of the steam with a small lift of 
the valve, otherwise the pressure of the steam may creep up considerably 
before the valve lifts sufficiently to discharge the steam. It is money 
well expended to place on each boiler two safety valves, with one set to 
blow off about 5 pounds beyond the other safety valve. If one safety 
valve gets out of order from any cause, then the other surety valve acts 
as an emergency valve, and when it blows off it indicates that the other 
safety valve is out of order. 

HEATING FEED-WATER. 

105. Mix hot water with cold water and the result will be that the 
temperature of the hot water decreases, while the temperature of the cold 
water increases. The feed-water furnished to a steam boiler should be 
heated to the normal temperature to that of steam. If the feed-water 
is fed to the boiler at 60°, it has to be heated from this temperature to 
that of the steam before evaporation can commence, and this means not 
only more fuel, but a reduction in the capacity of the boiler. A feed- 
water heater, where the water can be heated by the exhaust steam, or 
heated by the chimney gases, should be used to heat the feed-water 
before it is injected into the boiler. Aside from the above the feed-water 
heater, if the water is heated hot enough, will tend to remove some of the 
foreign substances, which exist in the water. Nearly all waters contain 
foreign substances in greater or less degree, and when the water is 
evaporated, the impurities remain in the boiler, being removed by blow¬ 
ing-off and washing out the boiler, and the latter can not be done too 
well, and should always be done when the boiler is cold. 







52 


BOILER CONSTRUCTION. 


The presence of scale or sediment in a steam boiler results in the 
use of more fuel, cracking of the plates, and leads to boiler repairs and 
explosions. It is estimated that -^-inch of scale causes a loss of 13 
per cent cf fuel; | inch causes a loss of 38 per cent, and £ inch causes a 
loss of 60 per cent.. The Railway Master Mechanics’ Association of the 
United States estimates that the loss of fuel, extra repairs, etc., due 
to incrustation, amount to an average of $750 per annum for every loco¬ 
motive in the Middle and Western states. 

SEPARATOR. 

106. In power plants where steam pipes extend from the boiler to the 
engine, it is well to place on the main steam pipe a separator, which will 
separate the water (the steam which condenses) from the steam as the 
latter passes from the boiler to the engine, etc. It is well known that 
the steam pressure along a steam pipe of considerable length is not the 
same throughout its length. The steam pressure in that part of the 
steam pipe furthest from the boiler will be less than that part of the 
steam pipe adjoining the boiler. The cooling of the steam enroute 
through the steam pipe causes more or less water to be in the steam 
pipe; hence, the separator to separate the water from the steam. When 
making equation of pipes it is well to bear in mind the velocity of the 
steam. The velocity of the steam will depend upon the size of the pipes 
and the number of angles. The friction is more in evidence in a small 
pipe than in a large pipe. To figure the area of the pipe desired for the 
main steam pipe, caring for reasonable resistance, etc., the velocity of 
the steam should be taken at about 85 feet per second, or 5,100 feet per 
minute. 

Pipes are measured from their inside diameters, while flues are 
measured from their outside diameters. Pipes are less uniform in 
diameter than flues* and calculations in regard to pipes may be upset, due 
to the diameter varying from the nominal diameter. Each boiler should 
be equipped with a valve so as to cut the boiler off from tne main steam 
pipe to which all the boilers, if there are a battery of boilers, are con¬ 
nected. This will permit the boiler to be cut out of service to make 
repairs, inspections, etc., without taking the other boilers out of service. 
When several boilers compose a battery, the pressure of each boiler 
should be as near alike as practicable, and when a boiler has been cut 
out of service, its steam pressure should be the same as the steam 
pressure of the other boilers before the valve to the main steam pipe is 
opened. Since water is practically incompressible, it will, when traveling 
with the steam, differ little from that of a solid body of equal weight; 
hence, will impart its weight against the elbows, valves, etc., and cause 
what is termed a HAMMER KNOCK, which frequently ruptures a pipe. 
Turning steam too rapidly into cold pipes will also rupture them. 


LAYING OUT. 


53 


LAYING OUT 


LAYING OUT BY PARALLEL LINES 

DRAFTING TERMS 

INTRODUCTION. 

1. The process of making from a drawing of a form a pattern upon 
a flat surface is called a DEVELOPMENT of the surface. To read a 
drawing a common understanding of drafting must first be acquired. The 
front view or the geometrical projection of a boiler or other object on a 
plane perpendicular to the horizon is called an ELEVATION. An ele¬ 
vation may be called front, side, end or rear, according to the dimensions 
of the object, one of whose faces it represents. The representation of 
tne boiler as it would appear if cut by the horizontal plane is called the 
PLAN VIEW. The name applies equally as well to a top view as a hori¬ 
zontal section. The representation of the object as it should appear if 
cut in two-through any plane, vertical, horizontal or oblique-is called 
a SECTIONAL VIEW. Generally the view is made known, such as section 
AA, etc. 

A drawing made less than full size of the object, each part drawn in 
proportion, is called a SCALE DRAWING. A drawing of a part, said 
drawing being made full size or to a scale generally greater than used 
for the elevation, is called a DETAIL DRAWING. 

THE PERSPECTIVE DRAWING. 

2. The representation by a drawing made on a flat surface of solid 
objects or surfaces conceived of as not lying in that surface; the delinea¬ 
tion of objects as they appear to the eye, is called a PERSPECTIVE 
DRAWING. In perspective the eye is supposed to carry a definite point, 
called the POINT OF SIGHT, and the picture is supposed to be at right 



54 


LAYING OUT. 


angles to the line of vision in a plane called the PLANE OF DELINEA¬ 
TION. 



In Fig. 1 is shown a form of perspective drawing, though the dotted 
lines, which represent unseen edges, are usually omitted. The outline or 
contour of a vertical section through a line of work, showing actual or 
projected elevation and hollows, is called a PROFILE. 


PRINCIPLES OF LAYING OUT. 

TAKE-UP IN ROLLING. 

3. When a sheet is changed from a flat to a curved surface the 
sheet at one and the same time is subjected to both compression and 
stretching. Since one is directly the opposite of the other, it follows that 























LAYING OUT. 


55 


the foregoing must be considered when figuring the circumference for a 
given diameter. 



In Fig. 2 the distance a, b, and c are the same. However, curve the 
sheet as shown in Fig. 3, and the distance a inceases, while the distance 
c decreases. Every circle, regardless of the diameter, has 360°, and as 
will be noticed, while the distance a, b, and c, each have the same num¬ 
ber of degrees, each has a different diameter; hence, a different length. 




Since rolling the sheet causes the distance a to increase and the dis¬ 
tance c to decrease in proportion, there naturally is a point that is 
neutral - that is, does not lose or gain. Practice has brought out that for 
all practicable purposes this point can be considered as in the middle of 
the plate; hence, the expression, NEUTRAL DIAMETER, as shown in 
Fig. 3. 






























56 


LAYING OUT. 


With very heavy plate, 1J inches or more in thickness, a slight V 
opening, as shown in Fig. 4, will occur. This, however, is cared for by 
slightly beveling the sheets prior to rolling. This feature may not be con¬ 
sidered with commercial sizes, say \\ inches and less in thickness. 

THE CIRCUMFERENCE. 

4. Many boiler makers figure out the circumference by multiplying 
the constant 3.1416 by the inside diameter, and add three thickness of 
plate for the take-up in rolling. Others multiply the constant 3.1416 by 
the outside diameter, and take off three thickness of plate for the gain 
in rolling. Neither of these rules are well to use, the better practice being 
to multiply the constant 3.1416 by the neutral diameter, which is readily 
found by adding to the inside diameter of the structure one thickness of 
plate. 

The latter is the best as it is as near accurate as possible, and fur¬ 
ther, the circumference ascertained requires no additions or deductions, 
except such allowances as are made for a loose fit. 

A plate is naturally rough, and its thickness not uniform. The varia¬ 
tion in the thickness may be inappreciable, though every plate is thicker 
in the center than at its edge. This is due to the spring in the rolls; the 
greater the width of the sheet and the lighter the thickness, the greater 
the variations. 

5. The commercial sheet nowadays does not vary to the extent that 
a special rule is necessary in every case. With steam-tight work and a 
two or more course structure, the circumference for both the large and 
small courses may be figured out by multiplying the constant 3.1416 by 
the respective diameters. Then, to permit the courses to be readily con¬ 
nected together, a slight addition to the large course, or a small deduc¬ 
tion from the small course might be made, but made prior to spacing off 
the rivet holes in the girth seam. 

I-n structures, such as stacks, stand pipes, etc., a greater allowance 
may be made than in the foregoing case. The difference in the circum¬ 
ference of the large and the small courses, when said circumferences are 
found by the neutral diameter method, is 6.28 times the thickness of the 
plate, the assumption being that both plates are the same thickness. In 
steam-tight work this may be made as great as 6.5 and in stack work as 
great as 7 times the thickness of plate. 

For instance: Assuming the circumference for a structure to be 
191.6376 inches for the large course, J inch plate, the circumference for 
the small course may be found for a very tight fit by merely subtracting 
from 191.6376 inches, 6.28 x .5 = 3.1416 inches, thus 191.6376-3.1416 = 188.- 
496 inches, say 188.5 inches. Ordinarily this would be written 191.6376- 
(6.28 x .5) =188.496 inches. 


LAYING OUT. 


57 


If a reasonable loose fit is desired, raise the constant from 6.28 to 
6.5, making a deduction of 6.5 x .5 = 3.25 inches, instead of 3.1416 inches. 
Or, if a very loose fit is desired raise the constant to 7, making a deduc¬ 
tion of 7 x .5 = 3.5 inches, instead of 3.1416 inches. 

DEVELOPING A SQUARE PIPE CUT OFF AT AN OBLIQUE ANGLE. 

6. The development of a plain pipe, as shown is Fig. 1, merely re¬ 
quires taking the distances a, b, c and d , and arranging them on the plate, 
so when it is cut out and formed it will be the shape desired. 



The half pattern, as shown in Fig. 5, is developed by drawing the 
three horizontal lines, making two of these equal in length to the distance 
b, Fig. 1, and the other line equal to the distance c. The vertical dis¬ 
tances a and b between the lines, Fig. 5, are made to correspond to the 
corresponding distances, Fig. 1. The distance c, Fig. 1, represents the 



























58 


LAYING OUT. 


number of degrees, or the cut off. The distance c and e, Fig. 1 and 5, 
should correspond, and the distance ft should correspond to the total of 
c and e. 

Note: The thickness of the plate is not considered, the whole aim, in 
the early part of the paper, being to bring out the principles of develop¬ 
ment. 


DEVELOPING A ROUND PIPE CUT OFF AT AN OBLIQUE ANGLE. 

7. In developing the square pipe, Fig. 1, but few measurements 
were required, but with the round pipe, as shown in Fig. 6, many lines 
are required. To secure data for laying out the pattern divide the semi¬ 
circle into any number of equal spaces-in this case six-and from the 
points 1 to 7 inclusive draw horizontal lines, as shown. Though the dis¬ 
tances between the horizontal lines are alike, they do not in the side ele¬ 
vation so appear, and, because of the round surface and the eye in a fixed 
position. 

The lines a and b, Fig. 6, as well as all the horizontal lines between 
them, are shown their true length. To develop the half pattern, Fig. 7, 
draw the stretchout outline M N of indefinite length, making the distance 
c between the outer lines equal to the distance between points 1 to 7 of 
the end elevation, Fig. 6. 




Fig. 6. 


Next divide the distance c, Fig. 7, into six equal spaces, each space 
numbered to correspond in length with the corresponding horizontal lines 
of the side elevation, Fig. 6. Following the locating of the points V to 7' 
inclusive, draw the irregular or camber line, and the half pattern, less 
laps, rivet holes, etc., is complete. However, it is customary to develop 
both halves at once, working from a common center line, which should be 
directly opposite the location of the seam or joint, and as the latter may 
be selected at different points, it follows that the center line selected may 
be any one of the horizontal lines, as shown in the side elevation, but in 






















LAYING OUT. 


59 


this case the top line is considered the center line, thus causing the seam 
to be located at the extreme bottom. 



DEVELOPING A SQUARE PIPE FITTING OVER THE RIDGE OF A 

ROOF. 

8. The development of a square pipe fitting over the ridge of a roof, 
etc., as shown in Pig. 8, is accomplished by drawing up the side elevation 




and the end view, and carefully noting the measurements, a, b, c and d. In 
this case the measurement d is one-half of c. 













































60 


LAYING OUT. 


To develop the half patterns, as shown in Fig. 9, draw the stretchout 
line M N of indefinite length, and then lay off the four equal spaces, mak¬ 
ing the distance between the lines equal to the measurement d of the end 
view, Fig. 8. From the points located on the stretchout line M N, Fig. 9, 



project horizontal lines, making the center line a equal to the distance o, 
Fig. 8. The other horizontal lines. Fig. 9, are made equal in length to the 
distance b, Fig. 8, and then connecting lines are drawn and the half pat¬ 
tern, less laps and rivet holes, is complete. 

DEVELOPING A ROUND PIPE FITTING OVER THE RIDGE OF A 

ROOF. 

9. The developing of the surface of a round pipe fitting over the 
ridge of a roof, etc., as shown in Fig. 10, is similar to developing the 
square pipe as described in Art. 7. The side and end elevation are drawn 
as usual, after which the semi-circle, and eievatlon, is divided into any 






















LAYING OUT. 


61 


number of equal spaces-in this instance divided into six spaces; tbe 
points are numbered from 1 to 7 inclusive. 



From the points, end elevation, project horizontal lines to the side ele¬ 
vation, as shown. The distance a is the over-all distance, though all the 
horizontal lines are shown their true length. 



To develop the half pattern, Fig. 11, draw the stretchout line M N of 
indefinite length, making the distance b equal to one-half of the circumfer 
ence of the pipe, or equal to the distance 1 to 7, Fig. 10. Then divide the 
d' itance b, Fig. 11, into six equal spaces, which will correspond with the 
spaces in the side elevation, Fig. 10. From the points located on the 











































62 


LAYING OUT. 


<r> 



Fig. 12. 
















LAYING OUT. 


63 


stretchout, Fig. 11, horizontal lines are projected and made to correspond 
in length to corresponding lines of the side elevation, Fig. 10. The ir¬ 
regular line, Fig. 11, from 1 to 7 is then drawn and the half pattern, less 
laps and rivet holes, is complete. 

THE JOINT BETWEEN TWO PIPES OF DIFFERENT DIAMETERS 
INTERSECTING AT OTHER THAN RIGHT ANGLE. 

10. To develop the branch pipe, as shown in Fig. 12, draw the end 
and side elevations. The end elevation - that is, the hole of the end ele¬ 
vation, must be developd from data procured from the side elevation. 


At 


—Z —J-V 


% 

Fig. 14. 

First draw the outline, side elevation, and then the circle. Divide the 
circle into any number of equal spaces - in this case four spaces, numbered 
1 to 5. From these points extend slant lines, as shown. 

To develop the elliptical hole, as shown in the end elevation, draw the 
semi-circle, dividing it off into the same number of equal spaces as the 
semi-circle, side elevation. From the points of the semi-circle, side eleva¬ 
tion, project horizontal lines intersecting the vertical lines projected from 
the points of the semi-circle of the end elevation. Through the points 1, 
2, etc., end elevation, draw the shape of the hole. The vertical lines in ad¬ 
dition also serve to develop the irregular curve from a to 5, etc., side ele¬ 
vation. This is accomplished by projecting the vertical lines from the 
semi-circle until they intersect the lower circle. From these points hori¬ 
zontal lines are projected to the side elevation and to intersect the slant 
lines, creating the points o, b, c, d and e. 

The half pattern, Fig. 13, is developed by first drawing the stretchout 
line M N, making the distance 1 equal to one-half of the circumference of 
the branch pipe. The distance f should be divided into four equal spaces, 
numbered from 1 to 5, all to correspond to the spaces and the numerals of 
the semi-circle of the side elevation. From the points on the stretchout 















64 


LAYING OUT. 


line M N project horizontal lines to correspond in length with the slant 
lines of the branch pipe, thus locating the points, a, b, c, d and e, Fig. 13. 
Draw connecting lines and the pattern, less laps and rivet holes, is com¬ 



plete. To develop the hole in the large cylinder course, draw the stretch¬ 
out line M N, Fig. 14, and set off the distance from a to c equal to corre¬ 
sponding distance, Fig. 12. Also make the distances g, h, i and j, Figs. 
12 and 14 to correspond. The distances Jc and l are then taken from the 



Fig. 16. 

end elevation, Fig. 12, and laid off, as shown in Fig. 14, thus locating the 
points a to e inclusive. The irregular curve is then drawn and the hole 
developed. 







































LAYING OUT. 


65 


THE DEVELOPMENT OF A THREE-PIECE 90° ELBOW. 

11. To develop the patterns for a three-piece elbow, Fig. 15, draw 
the two quadrants as shown. Then divide the quadrant a b into four equal 
spaces, and from the newly found points draw the dotted lines to the apex 
c. There is no need of dividing the quadrant d e as the equal spaces on it 
are located when the dotted lines are drawn. 

From the points a d project horizontal lines until they intersect the 
dotted line c /. From the points b c project vertical lines until they inter¬ 
sect the dotted line c g. Then draw the connecting 45° slant lines as 
shown. The semi-circle is next drawn and divided into any number of 
equal spaces-in this instance six spaces, numbered from 1 to 7. From 
these points project horizontal lines to the line c /, and then slant lines, 
all parallel, to line c g, and then vertical line to line b c. Usually the 
data is all taken from one section, as will be hereinafter explained. 

12. To develop the pattern, Fig. 16, which is the center course, draw 
the stretchout line M N, spacing off on it twelve equal spaces; said spaces 
to correspond with the spaces of the semi-circle, Fig. 14. Then from the 
newly found points, Fig. 15, which are numbered 1 to 7, project vertical 
lines on each side of the stretchout line M N, and make their lengths to 
correspond with the corresponding lengths of the slant lines of the center 
course Fig. 14. Then draw the irregular curve, Fig. 16, through the points 
1 to 7. and the pattern, less laps and rivet holes, is complete. It will be 
noticed that by cutting the pattern, Fig. 16, in two - that is, through the 
line M N, the end pieces are obtained, or in other words, two full patterns, 
one cut as described, will give the three pieces for the elbow, Fig. 15. 


LAYING OUT A DOME. 

13. When laying out the dome of a boiler the thickness of the ma¬ 
terial must be considered, and contrary to the general rule the measure¬ 
ments must be taken from the lengths of the lines determined by the 



Fig. 17. 


inside diameter-not by the lines de- 
termined by the neutral diameter as 
is the general rule. In Fig. 17 is 
shown the dome and a portion of the 
cylinder to which it is to be attached. 
Probably every boiler maker recalls 
flanging a dome or similar structure 
to strictly the flange marks, yet the 
dome would not fit the cylinder or 
shell-would have an opening at the 
top, said opening gradually tapering 
off to nothing at the sides, or else 
the dome would rock. 

The trouble, however, in such cases 





























66 


LAYING OUT. 


is due to the flange marks not being properly placed-that is to say, 
the dome was incorrectly laid out. The flanging of the dome to the 
supposed correct line, and then the alteration thereof, not only requires 
the flanging of the dome to be done over in part, but requires the metal 
to be unduly worked, which adds nothing to its strength. 

14. To correctly lay out the dome-that is, correct as far as prac¬ 
tical purposes is concerned, draw up the profile as shown in Fig. 17. 
Then draw the inside quadrant as shown, after which divide it into as 
many equal spaces as desired-any amount-in this case three equal 
spaces, numbered from 1 to 4 inclusive. From these points and through 
them project vertical lines intersecting the horizontal line AA and the 
top portion of the cylinder, which is indicated by a part of a circle; the 
lengths of the lines drawn are indicated by the letters a, b, c and d. 

15. The quadrant is not drawn to the neutral thickness of the plate 
for the reason that the flange marks, which should be on the inside of 
the dome body, want to be located so that when the dome is flanged it 
will fit the shell, except minor irregularities. If the quadrant is drawn 
to the neutral thickness of the plate, then line a will be the same length 
as line a', but lines b', c' and d' will not be the same as lines b, c and d - 
in fact the line d ' will be the distance e greater in length than line d, 
and just that amount out. 



16. To lay out the pattern, Fig. 18, draw the stretchout line AA of 
indefinite length, after which step off on it the equal spaces as shown, 
making the space between points 1 and 2 equal to one of the equal spaces 
of the quadrant to the neutral thickness of the plate, as shown in Fig. 

17. Special attention is directed that the spaces are taken from the 
quadrant drawn to the neutral thickness of the plate. In practice this 
is not done; the general rule is to ascertain the circumference by mul¬ 
tiplying the constant 3.1416 by the neutral diameter, and then divide 
the product by the number of spaces. The whole object of drawing the 
quadrant to the neutral thickness of the plate and placing in lines b\ c ■ 
and a' is to make clear why the dome does not properly fit. , 

Following the spacing of the points on the line AA, Fig. 18, project 
from the newly found points and at right angles (90°) to the line AA 




















LAYING OUT. 


67 


the vertical lines a, b, c and d, and make their lengths to correspond 
with the corresponding lines, Fig. 17. Points 1', 2 r , S' and !/, Fig. 18, 
are found by this method of procedure. Attention is directed to the 
dotted lines on part of the pattern, Fig. 18. These lines indicate how 
the dome would be laid out had the measurements a', b', c' and d' been 
used instead of the measurements a, ft, c and d. This readily illustrates 
the remarks in Art. 13. Following the locating of all the points (when 
one quarter has been developed the data for the other quarters has also 
been secured) then add for the flange and the lap, installing the cen¬ 
ter punch marks for the rivet holes and the pattern is complete. 



LAYING OUT A FLARING TRANSITION PIECE. 

17. The transition piece as shown in Fig. 19 is of that shape, in 
this instance, that it can be laid out by the parallel line method. How¬ 
ever, if the minor diameter of the oblong end and the diameter of the 
round end do not agree, then the pattern should and can be best laid 
out by triangulation. 

18. The respective end views are shown, though all the data could 
be secured from the side elevation and the one-half of the end elevation. 
To insure a clear understanding both end views are shown, it being ex¬ 
pected that after the principles of laying out are required that the unnec¬ 
essary parts as described will not be drawn up and used. 

19. First draw up the side and end elevations, after which divide the 
quadrants into any number of equal spaces - in this case three equal 
spaces, numbered from 1 to 4 inclusive. Then the dotted lines as shown 
in the side elevation are drawn in, said lines to be parallel to the solid 
line from a to b. Next draw the slant line, which is at right angles to the 
line a to b, to the point c. The horizontal line from c to d marks what 
might be termed the division point between the upper and lower half of 
the transition piece. The lower half, as will be noted, is merely one-half 
of a round cylinder, and the method of laying out, since it has already 































68 


LAYING OUT. 


been described (see Art. 4), need not be taken up at present. Tbe upper 
half, however, is of that character that it affords an opportunity to set 
forth very important features of a general character, which should be well 
understood as they are liable to present themselves in whole or in part 
from time to time. 

The next step is to take the distances e, / and g of the end elevations, 
and using as starting points where the slant lines are intersected by the 



v 


Fig. 20. 


solid slant lines, lay off the respective distances on the dotted slant lines 
which are parallel to the solid slant line a to b, thereby creating the 
points r to 4' inclusive, side elevation. Though the drawing does not 
clearly indicate that the line V to 4' is irregular, it is nevertheless. 

20. To develop the pattern, Fig. 20, make the line a to b equal to the 
line a to b, Fig. 19. Then draw the vertical parallel lines as shown in Fig. 
20, making the distances e, / and g to correspond to the corresponding 
distances of Fig. 19. Next, draw the horizontal line from c to c, making 
the distance a to 1\ Fig. 20, equal to the distance a to V Fig. 19. The 
length of the dotted lines, Fig. 19, which are parallel to the solid line 
a to b, are taken and are located in regards to line cc, Fig. 20, as they 
are in the side elevation in regards to the slant line, which is at right 
angles to the line a to b. 

The wedge-shaped section, Fig. 20, is the same as the wedge-shaped 
part shown in the side elevation; the distance h, Fig. 20, to correspond 
with the distance h, Fig. 19. The irregular lines are then drawn, and 
the development, less laps and location of rivet holes, is complete. 

















LAYING OUT. 


69 


TO DEVELOP A TWO-PIECE 90° ELBOW. 

21. To develop the patterns of a two-piece 90° elbow as shown in 



Fig. 21, requires the use of the same principles as used to develop the pat¬ 
terns for the three-piece elbow as described in Arts. 11 and 12. First 



draw up the profile, drawing the semi-circle as shown and dividing it off 
as in former problems. Attention is directed to the miter line, a to b. 





























70 


LAYING OUT. 


With the ordinary elbow the rivet holes are located on the miter line, 
but with a two-piece 90° elbow this is not practicable, therefore the pat¬ 
terns are laid out so that the line of rivet holes are not parallel with line 
a to b. The full lap on section A is allowed at the top (distance a to c), 
while the full lap of section B is allowed at the bottom (distance b to d), 
and the foregoing will be readily seen in the patterns, Figs. 22 and 23. 

22. To develop the patterns, Figs. 22 and 23, draw the stretchout lines 
M N, and then figure out the respective circumferences. The circumfer¬ 
ence c of section A, Fig. 22, should be about 6i times the thickness of the 
plate greater in length than circumference / of Fig. 23. By this it will be 
noted that section A is the large course and section B the small course. 
The manner of developing the pattern up to the center lines g and g\ Figs. 
22 and 23, is the same as developing the pattern, Fig. 16, as described in 
Art. 12. Refer to Fig. 23 and it will be noted that the distance between 
a and c is the same as a to c, Fig. 21, and that the lap allowed gradually 



reduces to nothing at b. Now, refer to Fig. 23 and it will be noted that the 
opposite is used - that is to say, the distance b to d is the same as b to d, 
Fig. 21, and that the lap gradually reduces to nothing at a. The rivet 
holes can be placed in section A, but it is not advisable to put them in sec¬ 
tion B. Though they are shown in the pattern, Fig. 23, they are merely 
shown in about their approximate location, and only shown to give a good 
idea of the location of the rivet holes in relation to the miter line g. 
Laps on the sides as shown are then added and patterns are complete. 
















LAYING OUT. 


71 


THE PATTERNS FOR A BIFURCATED PIPE, THE TWO ARMS BEING 
THE SAME DIAMETER AS THE MAIN PIPE, AND 
LEAVING IT AT THE SAME ANGLE. 

23. In Fig. 24 is shown an elevation of a bifurcated pipe, all arms be¬ 
ing of the same diameter. First, draw up the profile as in former prob¬ 
lems, drawing and dividing the semi-circle as heretofore described. The 
pattern for section A is similar to laying out a dome, which is described 



Fig. 24. 


in Art. 13 to 16 inclusive. The patterns for sections B and C are similar 
to the other patterns that have been described, but the manner of con¬ 
necting the sections together is important. While it may he possible to 
put a line of rivet holes on the miter line a to b, Fig. 24, the chances are 
in the majority of cases the rivets could not be driven, therefore, section 








72 


LAYING OUT. 


B is usually allowed a lap, extending over on section C as shown by the 
line c to d, and the rivet holes are placed as indicated. 

24. The patterns. Figs. 25 and 26 are laid out in the usual way by 
first drawing the stretchout line M N, and laying off the circumferences, 



which are represented by the letters e and /. The holes, marked 1', 2' and 
S', Fig. 26, s,re to be marked from the holes of section B. While shown in 
the pattern, Fig. 26, they are merely so located so as to give an approxi¬ 
mate idea of their location in regards to line g . All other holes, except 



Jf, 2 and S, Fig. 25, and l’,^2' and S', Fig. 26, can be located as shown—that 
is, where the vertical lines intersect the respective lines g and g. Add 
laps, etc., and patterns are complete. 

































LAYING OUT. 


73 


DEVELOPING IRREGULAR FORMS 


PRINCIPLES OF TRIANGULATION. 

INTRODUCTION. 

25. The majority of irregular surfaces are developed by triangulation, 
tion Triangulation is also employed to develop some regular telescop¬ 
ing forms that can not be readily developed by the radial method. De¬ 
veloping by triangulation is not as difficult as it appears, for triangu¬ 
lation is none other than creating a series of triangles whereby to ob¬ 
tain the true lengths of all foreshortened lines. In Fig. 27 is shown a 
cylinder, and to develop the pattern it is only necessary to employ what 
is known as the parallel line method. 

However, Fig. 27 affords an excellent opportunity to explain develop¬ 
ing by triangulation. Ordinarily with the parallel line method the dis¬ 
tance between the line ab and cd, and also the circumference, is all that 
is required in order to develop the pattern. With triangulation more 
lines will be required, therefore, in order to develop the surface of Fig. 
27 by triangulation, step off on the quadrant a given number of equal 
spaces-any amount-and then erect the vertical solid lines as shown, 
also put in the dotted slanting lines. The latter (the dotted lines) are 
not the true lengths, and are called foreshortened lines. 

26. In Fig. 27 all the solid lines appear to the eye the same length, 
while the dotted lines do not appear the same length. The latter is due 
to looking at a curved surface, the eye in a fixed position. The dotted 
lines encircle the cylinder like the thread of a screw, or spiral form. In 
Fig. 27 the true lengths of the solid lines can be taken directly from the 
elevation, and the true lengths of the dotted lines (in this case, all the 
dotted lines will be the same length) may be found by drawing, as shown 
in Fig. 28, the vertical line equal in length to the distance between the 
lines ab and cd. At the bottom of the vertical line and at right angles to 
it, draw a horizontal line of indefinite length and make the space a' equal 
to one of the equal spaces of the quadrant, and then draw the dotted line 



74 


LAYING OUT. 


as shown in Pig. 28, which is the true elevation of all the foreshortened 
dotted lines as shown in the side elevation. 

To develop the pattern Fig. 29, by triangulation, draw the center solid 
line equal in length to any of the solid lines, side elevation, Fig. 27. Then 
set the dividers to one of the equal spaces of the quadrant, Fig. 27, and with 



Fig. 28. Fig. 27 Fig. 29. 


1, Fig. 27 as a center, draw arcs as shown. Then take the length of the 
dotted line, Fig. 28, and with 1', Fig. 29, as a center, draw arcs intersecting 
the arcs previously drawn, thereby creating point 2. With V as a center, 
arcs are drawn at the bottom as at the top, and with point 2, Fig. 29, as a 
center, and trammels set equal to one of the solid lines, Fig. 27, draw arcs 
intersecting arcs previously drawn, thereby creating point 2'. Continue 
in this manner until the entire pattern is developed. 

The amount of equal spaces in the quadrant, Fig. 27, is optional, 
though it is advisable not to make the spaces too great. “Too great,” in 
this respect, means the exercise of judgment, the closer the spaces are 
together, the more accurate the development, providing care has been 






































LAYING OUT. 


75 


exercised. On some classes of work about a six inch space would be 

feasible, while on other classes 
of work about a two-inch space 
should be employed. 

27. In Fig. 30 is shown a cyl¬ 
inder, the upper base represent¬ 
ed by the line ab, which, as will 
be noted, is not in the horizontal 
plane. The pattern for Fig. 30, 
like Fig. 27, would be developed 
by parallel line method; the 
semi-circle being divided off 
into equal spaces as shown. To 
secure data for developing the 
pattern by triangulation draw the 
solid and dotted lines, Fig. 30, 
as described in connection with 
Fig. 27; the only difference being 
that the solid lines, Fig. 30, vary 
in length, though the true 
lengths in Fig. 30; as in Fig. 27, 
are shown in the side elevation. 
The true lengths of the dotted 
lines must be ascertained, and 
due to the position of the upper 
base, the altitudes for the dotted 
line will not be the same. The 
respective solid lines, Fig. 30, 
will, however, serve as the alti¬ 
tudes for the diagram of tri¬ 
angles Fig. 31, and the space a' 
Fig. 31, is equal to one of the 
equal spaces of the semi-circle. 
Fig. 30. 

Since the upper base is not 
in the horizontal plane, it fol¬ 
lows that the length of the line 
ab, from 1 ' to 9 ' is greater than 
the length of the line cd from 1 to 9, therefore, the development as shown 
in Fig. 32 becomes a necessity in order to obtain proper spaces whereby 
to develop the pattern, Fig. 33. 



Fig. 30. 






























76 


LAYING OUT. 


To develop Fig. 32 draw the line at from V to 9' equal in length to 
the line at Fig. 30, from 1 to 9, after which make the distance c, Fig. 32, 



Fig. 31. 


equal to the distance between 5' and 6', Fig. 30, the distance d, Fig. 32, 
equal to the distance between 6' and 7', Fig. 30; the distance c, Fig. 32, 

equal to the distance between 
7' and 8', Fig. 30; and the dis¬ 
tance /, Fig. 32, equal to the 
distance between 8' and 9', Fig. 
30. The vertical lines Fig. 32 
are erected at right angles to 
the line at, and the points 2' to 
8' inclusive located by measur¬ 
ing off on the respective vertical 
lines between the horizontal 
centel line of the semi-eircle, and points 2 to 8 inclusive; the distance g, 
Fig. 30, affords a clear explanation of the method of procedure. After 
the points 2 ' to 8' are located, draw the irregular curve. 

To develop the pattern. Fig. 33, erect the vertical solid lines from 
















LAYING OUT. 


77 


5 to 5', said line to be equal in length to the corresponding solid line. 



Fig. 34. 


Fig. 30. Set the dividers equal to the space between 5 and 6, Fig. 30, and 
with 5, Fig. 33, as a center, draw arcs as shown. Then set trammels equal 
to the dotted line 5' and 6', Fig. 31, and with 5', Fig. 33, as a center point, 
describe arcs intersecting arcs previously drawn thereby creating pointy. 
Now set dividers equal to the space 5' to 6 ', Fig. 32, and with 5', Fig. 33, as 
a center, draw an arc. Next set trammels equal to the solid line from 
6 to 6 ', Fig. 30, and with 4 , Fig. 33, as a center, draw an arc intersecting 
the arc previously drawn, thereby creating point If. A like process is 
continued until the entire pattern is developed; the space between 5 to 4 * 
4 to 3, etc., Fig. 33, to correspond to the space of the semi-circle. Fig. 30; 
the spaces between 5' to 4 ', 4 ' to S', etc., Fig. 33, to correspond with the 
spaces Fig. 32. 

It is self-evident that the length of the irregular line from V to 9 ', 
Fig. 33, is greater than the horizontal line from 1 to 9 , therefore, the space 
bewteen 4' to 5' must necessarily be greater than the space between 4 to 
5. Further, the spaces between 1' to 2\ 2 ' to 3', etc., are all uniform, while 
the spaces between 1' to 2', 2 ' to S’, etc., are not uniform, though, in this 
case, the spaces between 1' to 2', and 8 ' to 9' are alike, or in other words, 













78 


LAYING OUT. 


on the irregular curve there are four irregular spaces on each side of the 
center line. 

28. In Figs. 27 and 30 all the side lines in the side elevation were 
drawn in their true lengths hut in the frustrum of the cone as shown in 



Fig. 33 some of the solid lines, like the dotted lines, are foreshortened 
lines, which is due to the form being telescoping; the only solid lines in 
Fig. 34 that are shown in their true lengths are the lines from 1 to 1 \ 

To secure data for developing the pattern. Fig. 34, draw the two quad¬ 
rants as shown; step off equal spaces-any amount-each quadrant to be 
divided into the same number of equal spaces. Draw in the dotted and 
solid lines numbering them from 1 to 5 inclusive and from 1 ' to 5' inclu¬ 
sive. Then locate the corresponding points on the upper base line and 
the lower base line, after which draw in the solid and dotted foreshort¬ 
ened lines as shown in the side elevation. 

Since Fig. 34 is the frustrum of the cone, the pattern, in the majority 
of cases, could be more readily developed by the radical method, than 
by triangulation. However, in this instance, the object is to explain tri¬ 
angulation; the problem being an easy one and only requiring two tri¬ 
angles, or in reality, merely the finding of the true lengths of the dotted 
lines, which, in this case, are all the same length. The length of the 































LAYING OUT. 


79 


dotted lines need not be ascertained for it can be taken directly from the 
side elevation; the solid line from 1 to V is the true length of all the solid 
lines. 

To find the true length of the triangle as shown in Fig. 35, make the 

height of the altitude equal to 
the altitude of the frustrum, and 
at the foot of the altitude, Fig. 
35, and at right angles to it, 
draw a horizontal line of indefi¬ 
nite length. Make the space a, 
Fig. 35, equal to the dotted line 
a , Fig. 34, and then draw the 
dotted slanting line or the hypo¬ 
tenuse as shown in Fig. 35, 
which is the true length of all 
the dotted lines shown fore¬ 
shortened in the side elevation. 

To develop the pattern, Fig. 36, draw the center line from 5 to 5 ' 
equal in length to the solid line, Fig. 35, from V to 1', or equal to the solid 
line, Fig. 34, V to 1\ Now set the dividers to one of the equal spaces of 
the small quadrant, Fig. 34, and with 5, Fig. 36, as a center, draw an arc. 
Set the trammels equal to the dotted hyotenuse, Fig. 35, and with 5 ', 
Fig. 36, as a center, draw an arc intersecting the arc previously drawn, 




thereby creating point 4. Next set the trammels equal to the solid line 
1 to 1', Fig. 35, and with 4, Fig. 36, as a center draw an arc 
Next, set dividers to one of the equal spaces of the large 
quadrant, Fig. 34, and with 5 ' Fig. 36, as a center, draw, an arc 
intersecting arc previously drawn, thereby creating point 4. The bal- 












80 


LAYING OUT. 


ance of the pattern can be developed in a like manner; the spaces at 
the top to correspond to the spaces of the small quadrant, Fig. 34, and the 
spaces at the bottom to correspond to the large spaces, Fig. 34; the dotted 
and solid lines, Fig. 36, to correspond to the dotted solid lines of the 
diagram of the triangles, Fig. 35. After the pattern has been developed 
to 1, and 1' draw in connecting lines, etc., and the pattern is complete. 

29. The frustrum of a cone, Fig. 37, is similar to Fig. 34, except that 



the upper base is not in the horizontal plane. This problem affords a 
further opportunity to demonstrate developing by triangulation. To 
secure data for developing the surface, Fig. 37, draw the upper semi- 
























































LAYING OUT. 


81 


circle, Fig. 37, and divide it off into any number of equal spaces; in this 
case the points located are numbered from 1 to 5 inclusive. Also, draw 
the semi-circle, Fig. 37, and divide it off into the same number of equal 
spaces as the semi-circle, Fig. 37; the points located are lettered a to e 
inclusive. 

From 1 to 5 inclusive, upper plan, extend vertical lines to the dotted 
horizontal lines, Fig. 37, thus locating points l f to 5' inclusive; also ex¬ 
tend from a to e, lower plan, vertical lines to the lower base of Fig. 37, 
thereby creating points a' to e' etc., on the solid lines as shown in Fig. 37; 
the solid lines not to extend above the slant line, or points 1" to 5" in¬ 
clusive. Then draw the dotted lines as shown. The foregoing describes 
how points 1" to 5", Fig. 37, are located on the slant line ff. 

The data for developing the irregular curve, lower plan, is pro¬ 
cured from Fig. 37. From points 1' to 5' inclusive, Fig. 37, project hori- 
zontl lines through points 1" to 5" inclusive, Fig. 37, from the outer line 
to the center line 8 ' to c', and using g, upper plan, as a center, draw arcs, 
said arcs to intersect the radical lines extending from the apex g, upper 
plan, thereby locating points 6, 7, 8, 9, and 10; the vertical height be¬ 
tween the horizontal line gg and point 6, upper plan, being equal to the 
distance between the horizontal line g f g’ and point 6', Fig. 37. Points 7' 
8', 9 ', and 10', lower plan, are located in a like manner, after which the 
solid lines and dotted lines are drawn as shown. 

To develop the triangles, Figs. 38 and 39, make the altitudes A to D 
inclusive equal to the respective distances as shown in Fig. 37. As will 
be seen each dotted line has a different altitude; the same altitudes in 



some cases being the same for both the dotted and solid lines, though the 
hypotenuse of the triangles, Figs. 38 to 39 is not the same. The bases 
of the triangles which are taken from Fig. 37, make a difference in the 
lengths of the dotted and solid lines. The length of the dotted line, Fig. 
37, from 5" to c' is found by drawing the altitude A, Fig. 38, equal to the 
altitude A, Fig. 37, and making the base h, of the triangle, Fig. 39, 
equal to the dotted line h, Fig. 37, lower plan. The base of the balance of 
the triangles Figs. 38 and 39 are constructed in a like manner by taking 

















82 


LAYING OUT. 


the respective lengths of the dotted and solid lines from Fig. 37, lower 
plan. 


The foresnortened dotted and solid lines in both the side elevation, 
and the lower plan, Fig. 37, are shown so as to trace clearly the method 
of procedure; the dotted and solid lines, Fig. 37, side elevation, indicat¬ 
ing the points from 
which the altitudes 
are secured. The 
dotted and solid 
lines, lower plan, 
are none other than 
the base of the tri¬ 
angles ; the two 
foregoing when 
properly brought to¬ 
gether afford an 
opportunity to ascertain the hypotenuse of the triangle, which is the true 
length of the foreshortened lines. Before the pattern, Fig. 40, can be 
developed a development of the upper base must be made, therefore, 
Lnes at right angles to the slant line, //, are drawn and the distances 
from the line // to points 6" to 10 " inclusive are located; said lengths of 
lines to correspond to corresponding distances, Fig. 37, lower plan. 



To develop the pattern, Fig. 40, draw the solid line from o to o' equal 
in length to the solid line o to o', Fig. 37. Then set trammels equal to 
the dotted line, Fig. 39, altitude G, and using o, Fig. 40, as a center point, 
draw an arc. Set dividers equal to one of the equal spaces of the large 
semi-circle, Fig. 37, and with o', Fig. 40, as a center draw an arc inter¬ 
secting the arc previously drawn, thereby creating point a. With a, 


o 



Fig. 40, as a center and with the trammels set equal to the solid line, 
Fig. 38, altitude G, draw an arc. Then set dividers equal to the space 

















LAYING OUT. 


83 


o to 6", Fig. 37, and with o, Fig. 40, as a center, draw an arc, intersect¬ 
ing the arc previously drawn, thereby creating point 6'. 

The balance of the pattern is developed in like manner; the length 
of the respective dotted and solid lines being taken from Figs. 38 to 39; 
the spaces from a to b, b to c, etc., Fig. 40, to be taken from Fig. 37, 
lower plan; and the spaces 6' to 7', 7' to 8', etc., to be taken from the 
spaces 6" to 7", 7" to 8", etc.. Fig. 37, side elevation. After all the points 
are located draw in the connecting lines thereby delevoping the half 
pattern as shown. 


TO DEVELOP THE PATTERNS FOR A TRANSITION PIECE BE¬ 
TWEEN A RECTANGULAR OPENING AND A ROUND PIPE. 

30. The illustration in Fig. 41 represents a fitting or hood used 
by sheet metal workers. The bases for fans, ventilator stacks and 
smoke hoods are made in this form. Examination of the side eleva¬ 
tion shows that the lower base is in part parallel with the upper base; 
the balance of the lower base being obliquely inclined. 

Considering the foregoing, it will be seen that Fig. 41 really repre¬ 
sents on one side of the central vertical construction line one-half of 
the side elevation of a transition piece where the upper and lower bases 
are both in the same plane, such as would be the case if the transi¬ 
tion piece was to be attached to a flat roof, while in the other half of 
the side elevation the upper and lower base are not in the same plane, 
such as would be the case if the transition piece was attached to a 
slant roof. 

The principles of securing the data and developing the patterns 
are, however, the same, and accordingly Fig. 41 is assumed to be a 
transition so located on the roof that part of the lower base is attached 
to a flat roof, and the balance to a slant roof. The plan and elevation 
are first drawn as shown by the full lines, after which the semi-circle 
in the plan view is divideded into any number of equal spaces-in this 
case there are six spaces. The points are numbered from 1 to 7 in¬ 
clusive. From these points lines are extended to the points 8 and 9, 
which lines will later be used for the base of the triangles. 

If both the upper and lower base were in the same plane, then 
only one altitude would be necessary, the altitude a, but due to the 
obliquely inclined base the second altitude is necessary, the altitude b. 
In the side elevation are shown a number of lines which have no bear¬ 
ing on the problem except to show how the dotted lines, shown in 
the plan view, appear in the side elevation. Examination will show 
that the lines are projected from the points in the plan view to the 
upper base line. From the newly found points on said lines the dotted 
lines of the side elevation, which are foreshortened lines, are drawn 


84 


LAYING OUT. 


to points 8' and 9'. If the transition piece was to be attached to a 
curved surface, then more than two altitudes would be required and 
the foreshortened lines, as shown in the side elevation, would be nec¬ 
essary in order to locate the points whereby to determine the altitudes. 

The diagram of triangles, Fig. 42, are found in the usual way, 
making the altitudes a and b equal to the corresponding heights of 




Fig. 41 


Fig. 41. On the base of the triangles, Fig. 42, space off the distances 
from 1 to 8, 2 to 8, 3 to 8, 4 to 8, 4 to 10, 4 to 9, 5 to 9, 6 to 9 and 7 
to 9 of the plan view of Fig. 41, the distances being located on the 
base line, then drawing the connected lines to the top of the triangle, 
as shown in Fig. 42. The length of the solid lines, numbered 11 and 
12 in Fig. 42, are merely shown for clearness sake, and need not be 















































LAYING OUT. 


85 


ascertained, as they represent the solid lines of the side elevation from 
V to 8' and 7' to 9'. These measurements can he taken from the side 
elevation. The dotted line, numbered 10 in Fig. 42, is the length of 
the foreshortened line from 4 to 10, Fig. 41. 

To develop the pattern, Fig. 43, make the line from 4 to 10 equal 



to the slant solid line No. 10 of the altitude a, Fig. 42. Then set divid¬ 
ers equal to the space from 1 to 2, Fig. 41, and using point 4, Fig. 43, 
as a center point draw arcs on each side. The pattern for one side 
only need be described, for the same movements are used for one side 
as used for the other, but care should be taken at all times to use the 
proper lines. 



Having located points 4 and 10, Fig. 43, take the length of the 
line from 10 to 8 of the plan view of Fig. 41, and using point 10 of 
Fig. 43 as a center point draw an arc. Then with point 4 as a center 
point - trammels set equal to the distance of the dotted slant line 4 
of the triangle, altitude a , draw an arc intersecting the arc previously 
drawn and creating point 8. 









86 


LAYING OUT. 


The trammels are then set equal in length to the dotted slant line 
3 of the triangle, altitude a, and using point 8, Fig. 43, as a center 
point an arc is drawn intersecting the small arc previously drawn at 
the top, thereby creating point 3. Then the length of the dotted slant 
line 2 of the triangle, altitude a, is taken, and using point 8, Fig. 43, 
as a center, an arc is drawn. The dividers being set equal to the space 
from 1 to 2 of the plan view. Fig. 41. The point 3* Fig. 43, is used 
as a center point and an arc is drawn intersecting the arc previously 
drawn, thereby creating point 2. The trammels are then set equal to 
the distance of the slant dotted line 1 of the triangle, altitude a , and 
using 8, Fig. 43, as a center point, an arc is drawn. Then with set 
dividers and with point 2 as a center point, a small arc is drawn to 



intersect the arc previously drawn, thereby creating point 1. Next, the 
distance from 8 to 11 of the plan view of Fig. 41 is taken, and using 
point 8, Fig. 43, as a center, draw an arc. 

Following this operation, the trammels are set equal to the length 
of the solid line 1' to 8' of the side elevation, and using point 1, Fig. 
43, as a center point, draw an arc intersecting the arc previously drawn, 
thereby creating the point 11. The other part of the pattern is de¬ 
veloped in a like manner, except the length of the lines are taken from 
the triangle, altitude b. Allow for laps and the half pattern is com¬ 
plete. 

TO DEVELOP THE PATTERN FOR A TAPER COURSE OR SLOPE 

COURSE. 

31. A transition piece, called a taper coursd or slope course, is 
shown in the illustration, Fig. 44. The majority of locomotive boilers 
are nowadays constructed with a tape course of eccentric shape. To 
develop the pattern the end and elevation are first drawn up as shown 

























LAYING OUT. 


87 


by the heavy lines, and then the semi-circles of the end view are divid¬ 
ed into any number of equal spaces-in this case six spaces - numbered 
from 1 to 7 inclusive and V to 7' inclusive. 

After the several points are located then the dotted and solid con¬ 
necting lines in the end view are drawn. The same lines are also 
shown in the side elevation in their foreshortened length and correct 
position on the taper course, but their presence there is not of neces¬ 
sity to secure the data for laying out the pattern. They are merely 
shown to set forth the underlying principles of triangulation. The 
mean measurement of the side elevation noted in this problem is the 
distance a, which should be the over-all distance from flange line to 
flange line; also, the distances b to c, and d to e. 

The next in order is to find the 
true lengths of the foreshortened 
lines of the side elevation, which 
is accomplished by the diagram 
of triangles as shown in Fig. 45. 
The altitude a is to be the same 
as the distance a, Fig. 44. Two 
diagrams of triangles are shown, 
both with the same altitude, and 
this is done merely for clearness 
sake. As will be noted in the 
diagrams of triangles, Fig. 45, 
the dotted lines and solid lines 
are separated. The length of the 
respective lines are often nearly 
the same, and which, when work¬ 
ing to a scale, make them ap¬ 
pear about the same length; 
hence, arranging the triangles in 
the manner indicated. 

To secure the true lengths of the foreshortened lines of the side 
elevation take the length of the dotted lines from 1 to 2', 2 to 3', 3 to 
4', 4 to 5', 5 to 6', and 6 to 7', and the length of the solid lines from 
2 to 2', 3 to 3', 4 to 4', 5 to 5', and 6 to 6', and space them off on the 
base of the respective triangles in Fig. 45, and then draw the slant 
dotted and solid lines to the top as shown. The solid line 1 in the 
triangle, Fig. 45, need not be found, as it can be taken from the side 
elevation from the points c to b. It is, however, shown to show how 
the foreshortened line from 1 to 1', end view. Fig. 44, is found. The 
same method also applies to the foreshortened line from 7 to 7'; its true 
length is the line d to e in the side elevation. 

To develop the pattern, Fig. 46, make the center line from 1 to 1' 

















88 


LAYING OUT. 


equal to the length of the line in the side elevation from b to c. Then 
take two pairs of dividers, setting one qual to the space 1 to 2 of the 
end view, and the other to the space V to 2' of the end view. From 
here on with this problem they will be referred to as the “small space” 
for the former and the “large space” for the latter. The development 
is as follows: 

With point 1 as a center, Fig. 46, and trammels set equal to the 
length of the dotted line 2', Fig. 45, draw an arc. Then, with dividers 
length to the solid line 4 of the triangles, Fig. 45, draw an arc, and 
then with point 3, Fig. 46, as a center, dividers set to large space, draw 
an arc intersecting the arc previously drawn, thereby creating point 
2'. With 2' as a center point and trammels set equal to the solid line 
of the diagram of triangles, draw an arc and then with point 1 as a 
center point and dividers set equal to the large space, draw an arc 
intersecting arc previously drawn, thereby creating point 2. 

Using point 2 as a center and trammels set equal in length to 
the dotted line 3' of Fig. 45, draw r an arc and then with point 2' as a 
center divider, set the small space, draw an arc intersecting the arc 
previously drawn, thereby creating point 3'. With point 3' as a center 
point, trammels set equal in length to the solid line of 3 of Fig. 45, 
draw an arc and then with point 2, Fig. 46, as a center point, dividers 
set to the large space, draw an arc intersecting arc previously drawn, 
thereby creating point 3. 

Using point 3 as a center, trammels set equal in length to the 
dotted line 4', Fig. 45, draw an arc and then with point 3', Fig. 46 as 
a center, draw an arc intersecting the arc previously drawn, thereby 
creating point 4'. Using 4' as a center point, trammels set equal in 
set the small space and with point 1', Fig. 46, as a center point, draw 
an arc intersecting the arc previously drawn, thereby creating point 4. 
The other points, 5, 6, 7, 5', 6' and 7', are found in a like manner by 
taking the lengths of the line from the diagram of triangles, Fig. 45, 
and stepping off the small and large spaces as described. Add for flange, 
etc., and pattern is complete. 

TO DEVELOP THE PATTERN FOR A FOUR-WAY BRANCH Y. 

32. In sheet metal work there are many cases of branch pipe work 
where it is desired to take from the main pipe, called by many a LEAD¬ 
ER, a number of branches. The manner of designing a fitting would, 
perhaps, by different parties be designed differently, there being so 
many varying conditions which must be taken into consideration. How¬ 
ever, the first consideration should be to make the design such that 
an easy flow for the contents of the pipe will be secured. The angles, 
bends, etc., should be such that the work can be put together readily. 

In the illustration, Fig. 47, is shown a four-way branch pipe. The 




LAYING OUT. 




Fig. 46 













90 


LAYING OUT. 


problem is chiefly introduced to show fully the principles of triangula¬ 
tion, showing the use of certain principles heretofore mentioned. The 
first consideration is to decide upon the diameter of the main pipe; 
the same being shown in the plan view. The height and the diameter 
of the branch is next decided, the assumption being that all the branches 
are the same height and diameter. 


z. 



The side elevation of one of the branches, as well as the plan view 
of it, can not be drawn as readily as in former problems. It is a case 
of working from one to the other to draw the side elevation and the 
plan view of the branch pipe. First, the circle in the plan view is 
divided into three equal parts-the distance from A to B represents 
one-third of the circle, and is so placed that the distance from A to 
C, and B to 0, are the same on each side of the horizontal construction 
line as shown in the plan view. Then, the semi-circle as shown in the 
side elevation is drawn and divided into a given number of equal spaces 










































































LAYING OUT. 


91 


-in this case four equal spaces, numbered from 1 to 5 inclusive. A 
similar semi-circle is drawn in the plan view, being divided into the 
same number of equal spaces as the semi-circle in the side elevation. 
The points found on the semi-circle of the side elevation are then ex¬ 
tended to the slant line from 1 to 5, and from said line are projected 
to the plan view. Then lines are projected from the points found on 
the semi-circle, plan view, to intersect the lines projected from the side 
elevation, thereby developing the elliptical hole as shown in the plan 
view, the points of intersection of the afroesaid lines being numbered 
1', 2', 3', 4' and 5'. 

Inspection of the respective view will reveal that the branches are 
not only connected to the main pipe, but are also connected to one 
another, thereby making three seams which all come together in the 
center, or at the point D as shown in the plan view. The outline of 
the seam is shown in the side elevation from C' to A' to D'. The dis¬ 
tance from C' to A', as will be noted, is the foreshortened length of 
the distance from € to A of the plan view. The distance A' to D' is 
the manner in which the seam, shown by the line from B to D of the 
plan view, appears in the side elevation. 

The line from A' to D' is irregular, being developed in the follow¬ 
ing manner: Draw the quadrant as shown in (a) and divide into two 
equal spaces to correspond with the number of spaces from 1' to 3' of 
the elliptical hole of the plan view. From point 6 of the view (a) 
project a line at right angles to the line B to D, thereby locating the 
point 6'. From 6' project a vertical line of indefinite length, after 
which draw the quadrant as shown in the view (b), dividing the quad¬ 
rant into two equal spaces to correspond with the equal spaces of plan 
(a), and then from the point 6" project a horizontal line to intersect 
the vertical line projected from point 6', thereby creating point 6"'. 
The irregular curve from A' to D' is then drawn, passing through the 
point 6"'. 

The foregoing completes the drawing of the outline of the plan 
and side elevation, and also locates the points for drawing the fore¬ 
shortened lines. The dotted and solid foreshortened lines are then 
drawn in both the side elevation and the plan view, and in this prob¬ 
lem the foreshortened lines are necessary in the side elevation in order 
to secure the altitudes. In other problems the foreshortened lines have 
been shown in the side elevation and not used in every case, it being 
explained that they were merely shown to set forth the underlying prin¬ 
ciples of triangulation. 

The altitudes a, Fig. 47, as will be noted, is the vertical height of 

the heavy line from 1 to D\ the altitude b the height of the dotted line 

from 2 to D'; the altitude c the height of the solid line from 2 to 6"'; 

the altitude d the height of the dotted line from 6"' to 3; the altitude 


92 


LAYING OUT. 


e the height of the solid line from 3 to A'; the altitude / the height of 
the dotted line from A' to 4, and the height of the solid line from 4 
to 8'; nd the altitude g the height of the dotted line from 8' to 5, and 
the height of the solid line from 5 to C'. 

In Fig. 48 the diagram of triangles are shown, the altitudes a, b, 
c, d , e, f and g being made to correspond with the corresponding alti¬ 
tudes of the side elevation. In drawing up the triangles as shown in 
Fig. 48, the greatest of care must he taken. The base of the triangle, 
altitude a, Fig. 48, is the length of the solid line from 1 to D of the 
plan view. The hypotenuse of the triangle, altitude of Fig. 45, should 



Fig. 48 


equal the length of the solid line from 1 to D' of the side elevation. 
The base of the triangle of altitude b of Fig. 48, should be made equal 
to the dotted line of the plan view from D to 2'; the base of the tri¬ 
angle of the altitude c of Fig. 48 should be made equal to the solid 
line of the plan view from 2' to 6'; the base of the triangle of altitude 
d of Fig. 48, should be made equal to the dotted line of the plan view 
3' to A, or from 3' to B; the bases of the triangles of the altitude / of 
Fig. 48 should be made equal in length to the dotted line from A to 
4', and the solid line from 4' to 8 of the plan view; and the bases of the 
triangles of altitude g of Fig. 48 should be made equal to the length 
of the dotted line from 8 to 5', and from the horizontal construction 
line from 5' to C of the plan view. 




























LAYING OUT. 


93 


To develop the pattern, Fig. 49, make the line 1 to D’ equal to 
hypotenuse of the triangle of altitude a, Fig. 48, or equal to the solid 
line of the side elevation of the Fig. 47, from 1 to D\ Then set the 
dividers equal to the space from 1 to 2 of the semi-circle of the side 
elevation, and using point 1, Fig. 49, as a center, draw an arc. Next, 
set the trammels equal to the length of the slant line of altitude b, 
Fig. 48, and using point D 7 of Fig. 49 as a center, draw an arc inter¬ 
secting the arc previously drawn, thereby creating point 2. Then, using 
point 2 as a center point, trammels set equal in length to the solid slant 
line of altitude c, Fig. 48, draw an arc. With another pair of dividers 
set equal to one of the spaces of the quadrant of view (a). Fig. 47, 
and using point D 7 , Fig. 49, as a center, draw an arc intersecting the 
arc previously drawn, thereby creating point 6"'. With point 6"' as a 



center, trammels set equal in length to the dotted line of altitude d r 
Fig. 48, draw an arc. Then, with point 2, Fig. 49, as a center and 
dividers set to small space, draw an arc intersecting the arc previously 
drawn, thereby creating point 3. 

Using point 3 as a center, trammels set equal in length to the solid 
line of altitude e, Fig. 48, draw an arc, and then with point 6"' as a 
center, dividers set equal to the same space as between points D 7 and 
6"', draw an arc intersecting the arc previously drawn, thereby creat¬ 
ing point A. At this point a change takes place. The spaces from 1 
to 2, 2 to 3, etc., at the top of the pattern, Fig. 49, will be alike, while 
the spaces at the bottom of the pattern will not be alike throughout 
the whole development. The spaces from D' to 6" 7 and from 6" 7 to A 
were taken from the space in the plan view marked (a), while the 
balance of the spaces will be taken from another part of the plan view. 

Continuing the development, using point A, Fig. 49, as a center 








94 


LAYING OUT. 


point, trammels set equal in length to the dotted line of the altitude /, 
Fig. 48, draw an arc, and then with dividers set to the small space as 
used for the top of the pattern and using point 3, Fig. 3, as a center, 
draw an arc intersecting the arc previously drawn, thereby creating 
point 4. With point 4 as a center, trammels set equal to the solid line 
of altitude f, Fig. 48, draw an arc, and then using point A, Fig. 49, as 
a center, dividers set equal to the space from A to 8 of the plan view, 
draw an arc intersecting the arc previously drawn, thereby creating 
point 8. 

With point 8 as a center, trammels set equal to the dotted line of 
the altitude g, Fig. 48, draw an arc, and then using point 4 as a cen¬ 
ter, dividers set to small space, draw an arc intersecting the arc prev¬ 
iously drawn, thereby creating point 5, Fig. 49. Using point 5 as a 
center, trammels set equal to the solid line of altitude g, Fig. 48, draw 
an arc, and then with point 8 as a center, dividers set the same as last 
time, draw an arc intersecting the arc previously drawn, thereby creat¬ 
ing point C. Add laps, etc., and pattern for the branch pipe is com¬ 
plete. 


INTRODUCTION. 

33. Aside from the parallel line method and triangulation, certain 
patterns can be more readily and best be developed by the RADIAL 
LINE method. Its use, however, except approximate developments, is 
confined solely to forms that radiate from a common apex. A body 
may be telescoping in form, such as the taper course, Fig. 44, Art. 31, 
but the aforesaid taper course could not be readily developed by the 
radial line method due to the eccentric shape of the structure. 

TO DEVELOP THE PATTERN OF A CONE. 

34. The developing of the pattern of a cone is very simple. How¬ 
ever, it affords an opportunity to illustrate the foregoing remarks in 
Art. 33. The cone as shown in Fig. 50 radiates from the apex a; the 
distance b on each side of the vertical construction line being the same. 
In the half plan view, which is all that is needed to represent the ob¬ 
ject-and which is never drawn up when laying out the pattern for a 
cone-the points b' are the same distance from the center point o'. 
The cone, Fig. 51, therefore, explains fully the telescoping forms which 
can be developed by the radial line method. 

To develop the pattern of a cone as shown in Fig. 51, the measure¬ 
ments from a to b must be ascertained. This is usually done by draw¬ 
ing up a half side elevation. Then with trammels set equal to the dis¬ 
tance a to b, Fig. 50, the point a , Fig. 51, is used as a center to an arc 
drawn. The length of the arc, if the pattern is to be in one piece, must 
be great enough to permit the distance b to c, Fig. 51, to be equal to 
the circumference of the base of the cone of Fig. 50. The pattern, Fig. 


LAYING OUT. 


95 


51, is divided into four equal parts, each part representing one-fourth 
of the pattern; the distance b' to &' of Figs. 50 and 51 corresponding. 




Laps are allowed and rivet holes are installed and then the pattern is 
ready to be formed to the shape as shown in Fig. 50. It is usual, how- 



















96 


LAYING OUT. 


ever, to punch a small hole at a, Fig. 61, to permit the material to this 
point to gather. 


DEVELOPING A FRUSTUM. 

35. The frustum as shown in the illustration. Fig. 52, is that part 
of a cone contained between the base and the parallel plane AA to the 
base. The letters a, b, b f and c represent the same parts as the same 
letters do in Figs. 50 and 51. In fact the pattern, Fig. 53, is the same 
as Fig. 51, except the additional full line from A to A, which line in 
Fig. 53, represents the upper base AA of Fig. 52. The line AA is drawn 
by setting the trammels equal to the distance from a to A, Fig. 52, and 
then using a, Fig. 53, as a center draw an arc intersecting the lines 
from a to b, and a to c.. The circumference for the upper base need not 
be computed and laid off on the arc as is the case with the lower base. 
The circumference of the upper base at the points AA, where the arc 
intersects the lines as shown. 



TO DEVELOP THE PATTERN FOR A TELESCOPING TRANSITION 
PIECE WITH UPPER BASE OBLIQUELY INCLINED. 

36. The transition piece with the upper base obliquely inclined as 
shown in the illustration. Fig. 54, is the same as the frustum in all 
respects exeept the upper base represented by the letters AA'. The 
development of the pattern, Fig. 55, is more difficult than the develop¬ 
ment of the pattern, Fig. 53, requiring more data to be secured from 
the side elevation; to obtain the same the plan view must necessarily 
be drawn. 











LAYING OUT. 


97 


The outline of the pattern, Fig. 55, is developed in the usual way- 
that is, to set the trammels equal to the distance from a to &, Fig. 54, 
and then using a, Fig. 55, as a center, draw an arc, laying off on the 
arc the distance & to c equal to the circumference of the base of the 
body. 

The irregular line AA', Fig. 55, representing the obliquely inclined 
base of Fig. 54, can not be developed as easily as the line AA of Fig. 
53. First, it is necessary to divide the semi-circle of the plan view. 
Fig. 54, into a given number of equal parts - in this case four equal 
parts. From the points found on the semi-circle lines are projected to 
the lower base line 6Z>, and from these points to the apex a. Now, where 
the slant lines projected from the points e, c and d intersect the slant 
line AA', project horizontal lines to the slant line a to &, thereby creat¬ 
ing the points 1, 2 and 3. 

Inspection of Fig. 54 will reveal that the lines from a to & are 
their true length, while the lines from a to c, and a to d and a to e 
are not their true length, being only their foreshortened length. Since 
these lines are only foreshortened lines the true lengths of the lines 
between the lower base and the obliquely inclined vase can only be 
found by projecting over the points 1, 2 and 3 the horizontal lines as 
shown; thus, making the distance V to & the true length of the fore¬ 
shortened line from 1 to c; the distance 2' to & the true length of the 
foreshortened line from 3 c. 

To develop the irregular line through points A, V, 2', 3' and A', 
Fig. 55, the arc from a to & must be divided off into the same number 
of equal spaces as are in the base of the transition piece, which is eight 
equal spaces. Lines are then projected from the points c\ d' and 
c' to the apex a, and the distance V to A' is made equal to the distance 
& to A', Fig. 54; the distance c to 3, Fig. 55, is made equal to the dis¬ 
tance from 6 to 3', Fig. 54; the distance from d' to 2, Fig. 55, is made 
equal to the distance from & to 2', Fig. 54; and the distance from e' 
to 1', Fig. 55, is made equal to the distance from & to 1', Fig. 54. Add 
laps, etc., and pattern is complete. 

TO DEVELOP THE PATTERN FOR A FOOT-TUB. 

37. In Fig. 56 is shown the side elevation and half plan view of 
a transition piece, such as a foot-tub. The principles of radial line de¬ 
velopment are shown more extensively in this problem, for the bases 
are elliptical, and the sides flaring uniformly. Should, however, the 
sides not flare uniformly then the pattern could not be developed ac 
curately by the radial line method, triangulation being the proper meth¬ 
od, unless accuracy was not essential, and then an approximate method 
would do. 

When describing in Art. 34 the securing of the data for developing 


98 


LAYING OUT. 




Fig. 56. 


Fig. 57. 













































LAYING OUT. 


99 


the pattern of the cone, it was stated that the plan view need not be 
drawn up, or in other words all the data could be secured from the side 
elevation. When describing in Art. 36 the securing of data for devel¬ 
oping of the transition piece with the upper base obliquely inclined, 
the need for both the plan view and the side elevation was apparent. 

To develop the quarter pattern, Fig. 57, of the transition piece as 
shown in Fig. 56, it is necessary in order to secure the data to draw 
the side elevation, a half plan view and a half end view. The side 
elevation, end view and plan view are drawn to the desired measure¬ 
ment, the radii a, a', b and b' showing how the plan view is drawn so 
that the sides will taper uniformly. The difference between the radii a 
and a' should be the same as the difference between b and b\ 

The plan view may be located as near or as far from the side ele¬ 
vation as desired, for this has no bearing on securing the data for lay¬ 
ing out the pattern. First, extend the line from 1 to 2 of the side ele¬ 
vation until it intersects the vertical line c. The vertical line c is pro¬ 
jected from the center point d of the radii b and b\ It happens in this 
case that the plan view is located from the side elevation just the prop¬ 
er distance to make the slant line from 1 to 2 intersect the vertical 
line c at the point d on the horizontal construction line of the plan 
view. 

Next, the slant from 3 to 4 is extended an indefinite length, and 
then from the center point c of the radii a and a' and at right angles 
to the vertical construction line, draw the horizontal line until it inter¬ 
sects the slant line from 3 to 4 at point /. To develop the pattern, Fig. 
57, set the trammels equal in length from points 3 to /, Fig. 56, and 
then using point /, Fig. 57, as a center draw an arc. With / as a cen¬ 
ter point, trammels set equal to the distance 4 to /, Fig. 57, draw another 
arc. 


Then the distance g to ft, of Fig. 56 is taken and transferred as 
shown in Fig. 57, after which the line from ft to / is drawn. The tram¬ 
mels are then set equal to the distance from 1 to d, Fig. 56, and using 
point ft, Fig. 57, an arc is drawn intersecting the line from ft to / at 
point d. Then, using point d as a center, trammels set as before, draw 
an arc of indefinite length. Still using point d as a center, but tram¬ 
mels changed to the distance from 2 to d, Fig. 56, draw an arc. The 
distance from ft to 1 of the plan view, Fig. 56, is then ascertained, and 
using ft, Fig. 57, as a center, draw an arc intersecting the arc at point 
1, and from this point draw the line to point d as shown. Add laps, 
etc., and pattern is complete. 

DEVELOPING AN ELLIPSE. 

38. In Art. 37 it was mentioned that the base of the transition piece 
was elliptical shape. An approximate method of laying such a base 


100 


LAYING OUT. 


was given, the same being accurate enough for general purposes. A 
more accurate method of developing an ellipse is shown in Fig. 58. 
First, the major and minor diameter must be known, and then using 
the po’nt a as a center, trammels set equal to the respective radii, draw 
two circles as shown. 

Thus, if the major diameter is 16 inches and the minor diameter is 
10 inches, then the radius c would be eight inches, and the radius b 
would be five inches. Following the drawing of the circles, divide each 
quarter into a given number of equal spaces-any amount-the inner 
and the outer circles to have the same number of spaces. 



Fig. 58. 


In Fig. 58 only one-fourth of the circle is divided, this being all 
that is necessary in order to demonstrate the method, it being under¬ 
stood that all quarters are alike. Then the points on the inner and 
outer quadrant are number; those on the outer quadrant from 1 to 5 
inclusive, and those on the inner quadrant from 1' to 5' inclusive. Lines 
from points 2, 3 and 4 are then projected down, while lines from points 
2', 3' and 4' are projected over to intersect the foregoing lines, there¬ 
by creating the points 2", 3" and 4". Then draw the irregular curve 
and the eliptical hole is developed. 



* 





















LAYING OUT. 


101 


TELESCOPING FORMS 


TO DEVELOP THE PATTERN FOR A TELESCOPING TRANSITION 
PIECE INTERSECTING A CYLINDER. 

39. In Fig. 59 is shown a telespocing transition piece where the 
upper is in the horizontal plane and lower base intersects a cylinder. 
The data for laying out the pattern can anly he secured after the side 
elevation is completed by projecting points, hereinafter to be mentioned, 
from the end view to the side elevation. The side elevation, except 
the irregular line from a to b, and the end elevation are first drawn. 
Then the semi-circles in both the side elevation and end view are drawn 
and divided into any number of equal spaces-in this case three equal 
spaces, numbered from 1 to 4 inclusive in the side elevation, and from 
1' to 4' in the end view. 

From these points lines are then projected to the horizontal line 
1 to 1 of the side elevation and 4' to 4' of the end view; hence to the 
points c and e'. From the points d and e, where the slant lines to the 
apex & intersect the arc of the end view, project horizonal lines to in¬ 
tersect the corresponding lines of the side elevation, thereby creating 
the points d ' and e also, project from point 4' a line to intersect the 
center construction line of the side elevation, thereby creating point 4". 
The irregular curve from b to d’ to e' to 4" is then drawn and the side 
elevation is complete. 

40. The developing of the pattern. Fig. 60, is very simple. The 
trammels are first set to the distance o' to 4', Fig. 59, end view, and 
then using point c', Fig. 60, as a center, draw an arc. The length of 
the arc, shown from 4' to 4', is made equal to one-half of the circum¬ 
ference (the illustration Fig. 60 is only a half pattern) and then the 
circumference divided off into the same number of equal spaces as the 
semi-circle in the end view, Fig. 59. 

The points so located are numbered 1', 2' and 3', and from these 
points lines are projected to the apex c. Next, the length of the fore¬ 
shortened line from c' to a' of the same view. With trammels set to 
this distance and with point c', Fig. 60, as a center point, draw an arc 
intersecting the line V to c', thereby creating point a'. 

The length of the foreshortened line from d to c', Fig. 59, is secured, 
the same being equal to the distance from c' to d” of the end elevation. 
Setting trammels to this length and using point c', Fig. 60, as a center 
point, draw an arc intersecting the line from 2' to c', thereby creating 
point d". The length of the foreshortened line from e to c' of the end 
elevation is then ascertained, the same being equal to the distance from 
c' to e". Then using point c', Fig. 60, as a center point draw an arc 



102 


LAYING OUT. 


intersecting the line from 3' to o', thereby creating the point e". Draw 
the irregular curve from cZ' to d " to e" to 4', adding for laps, etc., and 
half pattern is complete. 

41. The hole in the cylinder can also be developed by data obtained 
from Fig. 59. First draw the stretchout line MN, Fig. 61, making the 
distance between a to d, d to e, e to 4' to correspond to the correspond¬ 
ing distance of the side elevation, Fig. 59. From these points project 
above and below the stretchout line MN vertical lines of indefinite 
length. The distance a to a', Fig. 59, is then taken and set off on each 
side of the line MN as shown in Fig. 61. Next, the distance from d' 
to the center construction line of the side elevation, Fig. 59, is taken 
and set off on each side of the line MN, Fig. 61, as shown. The meas¬ 
urement from e' to the vertical center construction line. Fig. 59, is taken 



and set off on each side of the line MN, Fig. 61, as shown, after which 
the irregular line is drawn through the points a' to d’ to e' to 4', and 
the irregular hole as required in the cylinder is developed. 

TO DEVELOP THE PATTERN OF A TELESCOPING TRANSITION 
PIECE INTERSECTING A CYLINDER AT AN ANGLE. 

42. In Fig. 62 is shown a telescoping transition piece intersecting 
a round body at an oblique angle. The principles of this problem are 
found more or less in other problems, the chief feature being the de¬ 
velopment of the irregular line from a to b. The side elevation is first 
































LAYING OUT. 


103 


drawn, except the irregular line from a to &, and then the quadrant of 
the plan view is drawn. 

Next, the semi-circle as shown in the side elevation is drawn and 
divided into equal spaces-any amount-in this case four equal spaces, 
numbered from 1 to 5 inclusive. Lines are then drawn from the base 
of the telescoping transition piece to the apex c. From the points 2', 
3' and 4', lines are projected to the plan view. On these vertical lines 

the distances from 4 to 4', 3 to 3' and 2 to 2' are made to correspond 

to the corresponding distances in the side elevation. Lines are then 
drawn from points 2', 3' and 4' of the plan view to the apex c'. 

Inspection will reveal that the line from c to 3' of the plan view 

intersects the quadrant at d; the line from c' to 2' intersects the quad* 


6 



Fig. 61. 


Fig. 60. 


rant at e; and the line from c' to 4' intersects the quadrant at /. Then, 
from points d, e and / project lines to intersect the lines in the side 
elevation, thereby creating the points d\ ef and f. The irregular line 
from a to & through the points d’, e’ and f can then be drawn. 

43. To develop the pattern, Fig. 63, set the trammels equal to the 
distance c to &, Fig. 62, and then using point o", Fig. 63, as a center 
draw an arc of indefinite length. The arc is then divided into four 
equal spaces, making the distance from 6 to 2, 2 to 3, etc., of Fig. 63 
to correspond with the corresponding distances of the semi-circle of the 
side elevation. 











Fig. 62. 


104 


LAYING OUT. 









































































LAYING OUT. 


105 


Then the length of the foreshortened line from f to c of the side 
elevation is found, the same being equal to the distance from e to 
and then using point c", Fig. 63, as a center, draw an arc intersecting 
the line from 2 to c", thereby creating point The length of the fore¬ 
shortened line of the side elevation from e' to c is ascertained, the same 
being equal to the distance from c to e", and then using the point c", 



Fig. 63, as a center, draw an arc intersecting the line from 3 to c", 
thereby creating point e 

The length of the foreshortened line from d' to c, Fig. 62, is ascer¬ 
tained, the same being equal to the distance c to d", and then using 
point c", Fig. 63, as a center, draw an arc intersecting the line from 













































106 


LAYING OUT. 


4 to c", thereby creating point d”. The length of the line from a to c, 
side elevation, is then taken, and using point c" as a center, draw an 
arc intersecting the line from 5 to c", thereby creating point a. Fig. 
63. Add laps, etc., and the half pattern is complete. The development 
of the hole at (a) is not described. Having been described in a prior 
problem, coupled together with the fact that the lines show fully the 
movement no further remarks are necessary. 



TO DEVELOP THE PATTERN OF A TRANSITION PIECE HAVING 
BOTH THE UPPER AND THE LOWER BASES 
OBLIQUELY INCLINED. 

44. In Fig. 64 is shown a transition piece which has both the up¬ 
per and lower bases obliquely inclined. The slant line a to b represents 
the upper base and the slant line c to d the lower base. Since the tran¬ 
sition piece is uniformly telescoping, all the lines radiate from the 
apex e. First the side elevation is drawn and then the plan view. The 
semi-circle is all that is necessary in the latter. It is divided into any 
number of equal spaces-in this case into six equal spaces - numbered 
from 1 to 7 inclusive. 

Lines from these points are then projected to the dotted lower base 








LAYING OUT. 


107 


line as shown; hence, to the apex e. Where the lines intersect the up¬ 
per and lower bases the points 2,, 3,, 4,, 5,, 6,, and 23', 4', 5' and 
6' lines are projected to the slant line from the apex e to d, thereby 
creating the points 2 „, 3 „ , 4 „, 5 6 „ , and 2", 3", 4", 5" and 6". 

45. To develop the pattern, Fig. 65, set the trammels equal to the 
distance e to d, Fig. 64, using point e', Fig. 65, as a center, draw an 
arc. On this arc step off six equal spaces, making the spaces from 1 
to 7 to correspond with the corresponding spaces of the semi-circle, plan 
view, Fig. 64. Trammels are then set equal to the distance e to b, Fig. 
64, and using point e, Fig. 65, as a center, draw an arc intersecting the 
line from e' to 1, thereby creating point b. Trammels are then set equal 
to the distance e to c, Fig. 64, and using point e', Fig. 65, as a center, 
draw an arc intersecting the line from e' to 1 at c. 

The distances between 2, to 2', 3 f to 3', 4, to 4', 5, to 5' and 6, to 6' 
are foreshortened lines-their true lengths are found on the slant line 
from e to d. Thus, the distance from 2" to 2, is equal to the foreshort¬ 
ened line from 2' to 2,. The length of the other foreshortened lines 
are found in a like manner; the same being clearly indicated in the 
side elevation. 

To develop the irregular lines in the pattern, Fig. 65, from a to b, 
and c to 7, merely requires taking the length of the lines from the side 
elevation and transferring them to the corresponding lines in Fig. 65. 
Thus, the distance from e' to 2 „ is equal to the distance from e to 2 „ 
of Fig. 64; the distance from e' to 2", Fig. 65, is equal to the distance 
from e to 2", Fig. 64. This process is continued until the points 2 „, 3 „, 
4 „, 5 „, 6 „, and 2", 3", 4", 5" and 6" are located and then the irregu¬ 
lar lines as shown are drawn. Add laps, etc., and half pattern is com¬ 
plete. 


108 


MENSURATION. 


LINES. 


-B 


Fig. 1. 


A STRAIGHT LINE is one that does 
not alter its direction through its en¬ 
tire length, thus A to B, Fig. 1, is a 
straight line. 


Fig. 2. 


Lines which are an equal distance 
apart are PARALLEL LINES. See 
Fig. 2. 





A line changing its direction at 
every point is a CURVED LINE. See 
Fig. 3. 


A PERPENDICULAR LINE is one 
that is at right angles to another line. 
Thus, line AB is perpendicular to line 
CD, Fig 4. 


A VERTICAL LINE is a line that 
points towards the center of the earth. 
The line AB, Fig. 5, is a vertical line; 
also, in this case a perpendicular line. 









MENSURATION. 


109 



The illustration, see Fig. 6, bounded 
by a curved line, every point of which 
is the same distance from the center 
A, called the APEX, is a CIRCLE. The 
curved part of the circle from C to D 
is called an Arc; the shaded part is 
called a SEGMENT; the horizontal 
distance from C to D is a CHORD; the 
over-all distance E, passing through 
the center of the circle, is the DIAM¬ 
ETER; and the distance F is the 
RADIUS. 


Every circle, regardless of its diam¬ 
eter, has 360 parts, called DEGREES; 
each degree is subdivided into 60 parts, 
called MINUTES, and each minute is 
subdivided into 60 parts, called SEC¬ 
ONDS. The total distance around the 
circle is called the CIRCUMFERENCE. 
One-fourth of a circle is called a 
quadrant. 


A curved line as shown in Fig. 7 
is called an IRREGULAR CURVE; 
the illustration, Fig. 8 is called an 
OFFSET; also, a OGEE CURVE. 











110 


MENSURATION. 



An opening between two lines which 
intersect or meet is called an ANG-LE; 
the meeting point is called the VER¬ 
TEX. In Fig. 9 is shown a RIGHT 
ANGLE, so called as the vertical line 
is at 90° or at right angles to the hor¬ 
izontal line. 


In Fig. 10 is shown an ACUTE 
ANGLE, so called as the angle is less 
than a right angle. In Fig. 11 is shown 
an OBTUSE ANGLE, so called as the 
angle is more than a right angle. 


The BASE is the side of a plane 
figure on which the structure is sup¬ 
posed to stand. The line G to D, Fig. 
4, is the base line. 


Fig. 10. 



The ALTITUDE of a structure is 
its height, thus, the distance A to B, 
is the altitude of Fig. 4. 


A line touching a curve as does the 
line A to B, Fig. 12, is said to be 
TANGENT. 






MENSURATION. 


Ill 



Fig. 12. 


The SECANT is a line drawn from 
the center of a circle through one end 
of an arc, and terminated by a tangent 
drawn through the other end; the line 
A to C, Fig. 12, illustrates the same 


The SECTOR is a portion of the 
area of a circle included between two 
radii and an arc, as shown by the 
shaded part, Fig. 12. 


To find the area of a circle, square the diameter and multiply by 
the constant .7854. 

Example: What is the area of a circle whose diameter is 10 inches? 

Solution: 102 = 100; and 100 x .7854 x 78.54 sq. in. 

To find the circumference multiply the diameter by the constant 
3.1416. 

Example: What is the diameter of a circle whose diameter is 

10 inches? 

Solution: 10 x 3.1416 = 31.416 inches. 

When a circle is divided into two equal halfs, each half is called 
a SEMI-CIRCLE, and each half of the circumference is called the SEMI- 
CIRCUMFERENCE. A SPHERE, or a BALL, or a GLOBE, is a solid 
bounded by a uniformly curved surface, every point the same distance 
from the center. To find the area of a sphere, square the diameter and 
multiply the result by 3.1416. 





112 


MENSURATION. 


The circular ring is the area be¬ 
tween two concentric circles, being 
indicated by the shaded part, Fig. 13. 

Example: The diameter of the large 
circle is 10 inches and the diameter 
of the small circle is 5 inches, what 
is the area of the circular ring? 

Solution: The area of the large 

circle is: 102x .7854 = 78.54 sq., in.; the 
area of the small circle is 524-.7854= 
19.635 sq. in. The area of the circular 
ring is the difference between these 
areas, or 78.54-19.635=58.905 sq. in. 

Fig. 13. 

The area of a segment cannot be found exactly except by principles 
of trigonometry, but an approximate rule gives results close enough for 
practical purposes. 

Where: 

A = Area of segment in square inches. 

CD = Length of chord of the segment in inches. 

G= Height of the segment in inches. 

G8 2CD x G 

-L -= A 

2CD % 3 

Example: Assuming the distance from Cl to D, Fig. 6, to be 8 inches 
and the height G of the segment to be 2 inches, what is the area of the 
segment? 

Solution: 

8 16x2 

-+ 



16 


3 


= 11.166 sq. in. 






MENSURATION. 


113 



Fig. 15. 

— RIGHT-a* 


In Fig. 14 is shown a right angle 
triangle, the distance A to B being 
the base, the distance A to C the alti¬ 
tude, and the slant line from B to C 
is called the HYPOTENUSE. 

In drawings the lines are made 
various ways, each way indicating a 
certain practice. The very light full 
line. Fig. 15, is used mostly as a di¬ 
mension line. The dotted line is used 
to show parts hidden from view. The 
dash and two dot line is used to in¬ 
dicate the center lines; also, to indi¬ 
cate where a section has been taken 
when a sectional view is shown. The 
dash and one dot line is used mostly 
in projecting from one view to an¬ 
other, or to a dimension line. The 
heavy full line, made at least twice as 
thick as the light full lines, is used for 
shade lines. 


WRONGS 



Dimension lines should have at their 
ends arrow-heads, not too flaring, the 
right and the wrong way being shown 
in Fig. 16. 

The SCALENE TRIANGLE, Fig. 17, 
is so-called as none of its two sides 
are of equal length. 






















114 


MENSURATION. 


GEOMETRICAL FIGURES—POLYGONS. 

A triangle is a figure with three sides and three angles; a square 
is a four-sided figure, all the angles of which are right angles, and all 
sides equal. 


Triangle. 



Square. Pentagon. Hexagon. Heptagon. 

□ooo 



Octagon. 


Nonagon. Decagon. Undecagon. Dodecagon. 


Fig. 18. 


A pentagon is a plane figure with five sides and five angles; a hexagon 
with six sides and six angles; a heptagon with seven sides and seven 
angles; an octagon with eight sides and eight angles; a nonagon with 
nine sides and nine angles; a decagon with ten sides and ten angles; an 
undecagon with eleven sides and eleven angles; and a dodecagon with 
twelve sides and twelve angles, all as shown in Fig. 18. 




DECIMAL EQUIVALENTS. 


TABLE OF DECIMAL EQUIVA¬ 
LENTS 


OF EIGHTHS, SIXTEENTHS, THIRTY-SEC¬ 
ONDS AND SIXTY-FOURTHS 
OF AN INCH. 

TABLE I. 


A .015625 

& .03125 

& .046875 

* .0625 


II 

..515625 

u . 

.53125 

M . 

.546875 

. 

.5625 


A .078125 

& .09375 

A .109375 

£ .1250 


H.578125 

.59375 

If.609375 

| .6250 


A . 

.140625 

04 •••••••« 

^2 . 

.15625 

II . 

.171875 

. 

.1875 

If . 

.203125 

$2 .- 

.21875 

if .. 

.234375 

1 . 

.2500 

II . 

.265625 

A . 

.28125 

if . 

.296875 

.' 

.3125 

II . 

.328125 

II . 

.34375 

H . 

.359375 


.3750 


.. 

.640625 

U . 

.65625 

. 

.671875 

11 . 

.6875 

if . 

.703125 


.71875 

ii .. 

.734375 

i .7500 


||.765625 


52 . 

|| . 

.796875 

11 . 

.8125 

£f . 

.828125 

64 .. 

.84375 

5f . 

.859375 

64 . 

I . 

.8750 


If 

II 

*i 

* 


.390625 

.40625 

.421875 

.4375 


II 

II 

II 


.890625 

.90625 

.921875 

.9375 


II 

II 

II 

I 


.453125 

.46875 

.484375 

.5000 


.953125 

.96875 

.984375 

1.0000 









































































116 


FINDING DECIMALS. 


CONSTANT FOR FINDING DIAMETER AT 
BOTTOM OF THREAD. 

TABLE II. 


Threads 
per Inch 

u. s. 

Standard 

Constant 

V Thread 
Constant 

Threads 
per Inch 

u. s. 

Standard 

Constant 

Y Thread 
Constant 

64 

.02030 

.02706 

16 

.08119 

.10825 

60 

.02165 

.02887 

14 

.09279 

.12372 

56 

.02320 

.03093 

13 

.09993 

.13323 

50 

.02598 

.03464 

12 

.10825 

.14434 

48 

_ .02706 

.03608 

11 

.11809 

.15746 

44 

.02952 

.03936 

10 

.12990 

.17321 

40 

.03248 

.04330 

9 

.14434 

.19245 

36 

.03608 

.04811 

8 

.16238 

.21651 

32 

.04059 

.05413 

7 

.18558 

.24744 

30 

.04330 

.05773 

6 

.21651 

.28868 

28 

.04639 

.06186 

H 

.23619 

.31492 

26 

.04996 

.06662 

5 

.25981 

.34641 

24 

.05413 

.07217 

4J 

.28868 

.38490 

22 

.05905 

.07873 

4 

.32476 

.43301 

20 

.06495 

.08660 

3i 

.37115 

.49487 

18 \ 
1 

.07217 

1 

.09623 1 

1 

1 3 

1 1 

.43301 | 
1 

.57733 


C = Constant for number of threads per inch. 

D = Outside diameter. 

Di = Diameter at bottom of thread. 

D 1 = D—C. 

EXAMPLE. 

The outside diameter of U. S. S. screw thread is 2 
inches; 4£ threads per inch; find diameter at bottom of 
thread. 2 - .2886 = 1.7114 inches. 




















STEAM TABLES. 


117 


STEAM TABLES. 

TABLE III. 


Pressure in 
Pounds per 
Square Inch 
above vacuum 

Temperature 
in Degrees 
Fahrenheit 

Pressure in 
Pounds per 
Square Inch 
above vacuum 

Temperature 
in Degrees 
Fahrenheit 

2 

126.3 

104 

330.4 

4 

153.1 

108 

333.2 

6 

170.1 

110 

334.6 

8 

182.9 

114 

337.2 

10 

193.3 

118 

339.8 

12 

202. 

120 

341.1 

14 

209.6 

124 

343.5 

14.7 

212 

128 

345.9 

16 

216.3 

130 

347.1 

18 

222.4 

134 

349.5 

20 

228 

138 

351.7 

22 

233.1 

140 

352.9 

24 

237.8 

144 

355.1 

26 

242.2 

148 

357.2 

28 

246.4 

150 

358.3 

30 

250.3 

154 

360.3 

32 

254 

158 

362.4 

34 

257.5 

160 

363.4 

36 

260.9 

164 

365.4 

38 

264.1 

168 

367.3 

40 

267.1 

170 

368.3 

42 

270.1 

174 

370.2 

44 

272.9 

178 

372.1 

46 

275.7 

180 

373 

48 

278.3 

184 

374.8 

50 

280.9 

188 

376.6 

54 

285.7 

190 

377.4 

58 

290.3 

194 

379.2 

62 

294.7 

198 

380.9 























118 


STEAM TABLES. 


TABLE Ml—CONTINUED. 


Pressure in 
Pounds per 
Square Inch 
above vacuum 

Temperature 
in Degrees 
Fahrenheit 

Pressure in 
Pounds per 
Square Inch 
above vacuum 

Temperature 
in Degrees 
Fahrenheit 

1 

66 

298.8 

200 

381.7 

70 

302.7 

204 

383.4 

1 

74 

306.5 

208 

385.1 

78 

310.1 

| 

210 

385.9 

82 

313.5 

214 

387.5 

86 

316.8 | 

218 

389.1 

90 

320 1 

220 ' 

389.8 

94 

323.6 

224 

391.4 

98 

326.1 

228 

392.9 

100 

327.6 

230 1 

393.7 

234 

395.2 

274 

409.2 

238 

396.7 

278 

410.5 

240 

397.4 

280 

411.1 

244 

398.9 

284 

412.4 

248 

400.3 

288 

413.7 

250 

401 

290 

414.3 

254 

402.4 

' 294 

415.6 

258 

403.8 

298 

416.8 

260 

404.5 

I 300 

1 417.4 

264 

405.8 

350 

431.9 

268 

407.2 

400 

445.2 

270 

407.9 

1 

500 

1 

I 466.6 

1 


To reduce to guage pressures at sea level subtract 14.7 
from pressures given in the column. For instance, if the 
gauge pressure is 170 pounds, then the pressure in pounds 
per square inch above vacuum will be nearly 185 pounds. 
In altitudes above sea level, subtract pressures per square 
inch as given in Table No. IV. 

















BOILING POINT IN DEGREES. 


BOILING POINT IN DEGREES. 

TABLE IV. 


Boiling point in 
degrees, Fahrenheit 

Altitude above sea- 
level In feet 

Atmospheric pressure 
in lbs. per square inch 

184 

15,221 

8.19_ 

185 

14,649 

8.37 

186 

14,075 

8.56 

187 

13,498 

8.75 

188 

12,934 

8.94 

189 

12,367 

9.13 

190 

11,799 

9.33 

191 

11,243 

9.53 

192 

10,685 

9.74 

193 

10,127 

9.95 

194 

9,579 

10.16 

195 

9,031 

10.38 

196 

8,481 

10.60 

197 

7,932 

10.82 

198 

7,381 

11.05 

199 

6,843 

11.28 

200 

6,304 

11.52 

201 

5,764 

11.76 

202 

5,225 

12.01 

203 

4,697 

12.25 

204 

4,169 

12.51 

205 

3,642 

12.77 

206 

3,115 

13.03 

207 

2,589 

13.29 

208 

2,063 

13.57 

209 

1,539 

13.84 

210 

1,025 

14.12 

211 

512 

14.41 

212 1 

Sea-level 

14.70 
















120 


AREAS AND CIRCUMFERENCE OF CIRCLES. 


AREAS AND CIRCUMFERENCE OF CIRCLES. 

TABLE V. 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

s’* 

.0491 

.0002 

4J 

14.1372 

| 15.9043 

3*2 

.0982 

.0008 

41 

14.5299 

j 16.8002 


.1963 

.0031 

41 

14.9226 

17.7206 

I 

.3927 

.0123 

41 

15.3153 

| 18.6655 

* 

.5890 

.0276 

5 

15.7080 

! 19.6350 

S 

.7854 

.0491 

5i 

16.1007 

i 20.6290 

tk 

.9817 

.0767 

5i 

16.4934 

j 21.6476 

i 

1.1781 

.1104 

51 

16.8861 

1 22.6907 


1.3744 

.1503 

5S 

17.2788 

| 23.7583 

2 

1.5708 

.1963 

5§ 

17.6715 

24.8505 

t 9 s 

1.7671 

.2485 

51 

18.0642 

25.9673 

i 

1.9635 

.3068 

51 

18.4569 

j 27.1086 

» 

2.1598 

.3712 

6 

18.8496 

| 28.2744 

3 

2.3562 

.4418 

6i 

19.2423 

| 29.4648 

it 

2.5525 

.5185 

6| 

19.6350 

| 30.6797 

I 

2.7489 

.6013 

61 

20.0277 

i 31.9191 

If 

2.9452 

.6903 

6 \ 

20.4204 

33.1831 

1 

3.1416 

.7854 

61 

20.8131 

34.4717 

IS 

3.5343 

.9940 

61 

21.2058 

35.7848 

is 

3.9270 

1.2272 

61 

21.5985 

37.1224 

11 

4.3197 

1.4849 

7 

21.9912 

38.4846 

IS 

4.7124 

1.7671 

7| 

22.3839 

39.8713 

11 

5.1051 

2.0739 

7i 

22.7766 

41.2826 

11 

5.4978 

2.4053 


23.1693 

42.7184 

IS 

5.8905 | 

2.7612 

7i 

23.5620 

44.1787 

2 

6.2832 | 

3.1416 

7| 

23.9547 

45.6636 

21 

6.6759 ! 

3.5466 

71 

24.3474 

47.1731 

2i 

7.0686 | 

3.9761 

71 

24.7401 

48.7071 

21 

7.4613 | 

4.4301 

8 

25.1328 

50.2656 

2S 

7.8540 

4.9087 

8S 

25.5255 

51.8487 . 

21 

8.2467 

5.4119 

si 

25.9182 

53.4563 

21 

8.6394 

5.9396 

81 

26.3109 ! 

1 55.0884 

21 

9.0321 

6.4918 

8i 

26.7036 

56.7451 

3 

9.4248 

7.0686 

81 

27.0963 

58.4264 

3i 

9.8175 ! 

7.6699 

81 

27.4890 

60.1322 

3S 

10.2102 

8.2958 

81 

27.8817 

61.8625 

31 

10.6029 

8.9462 

9 

28.2744 

63.6174 

3S 

10.9956 

9.6211 

9i 

28.6671 

65.3968 

31 

11.3883 I 

10.3206 

9S 

29.0598 

67.2008 

31 

11.7810 j 

11.0447 

91 

29.4525 

69.0293 

31 

12.1737 

11.7933 

n 

29.8452 

70.8823 

4 

12.5664 | 

12.5664 

91 

30.2379 

72.7599 

4| 

12.9591 

13.3641 

91 

30.6306 

74.6621 

4S 

13.3518 1 

14.1863 

91 

31.0233 

76.589 

4| 

13.7445 i 

15.0330 

10 | 

31.4160 | 

78.540 

























AREAS AND CIRCUMFERENCE OF CIRCLES. 


121 


TABLE—CONTINUED. 


i 


Diam. 

Circum. 

Area. 

101 

31.8087 

80.516 

10| 

32.2014 

82.516 

101 

32.5941 

84.541 

lOi 

32.9868 

86.590 

10| 

33.3795 

88.664 

101 

33.7722 

90.763 

101 

34.1649 

92.886 

11 

34.5576 

95.033 

111 

34.9503 

97.205 

111 

35.3430 

99.402 

111 

35.7357 

101.623 

111 

36.1284 

103.869 

111 

36.5211 

106.139 

111 

36.9138 

108.434 

111 

37.3065 

110.754 

12 

37.6992 

113.098 

121 

38.0919 

115.466 

121 

38.4846 

117.859 

121 

38.8773 

120.277 

121 

39.2700 

122.719 

12| 

39.6627 

125.185 

121 

40.0554 

127.677 

121 

40.4481 

130.192 

13 

40.8408 

132.733 

131 

‘ 41.2335 

135.297 

131 

41.6262 

137.887 

13| 

42.0189 

140.501 

131 

42.4116 

143.139 

13| 

42.8043 

145.802 

131 

43.1970 

148.490 

131 

43.5897 

151.202 

14 

43.9824 

153.938 

141 

44.3751 

156.700 

141 

44.7678 

159.485 

141 

45.1605 

162.296 

141 

45.5532 

165.130 

141 

45.9459 

167.990 

141 

46.3386 

170.874 

141 

46.7313 

173.782 

15 

47.1240 

176.715 

151 

47.5167 

179.673 

151 

47.9094 

182.655 

15| 

48.3021 

185.661 

151 

48.6948 

188.692 

151 

49.0875 

191.748 


Diam. 

Circum. 

Area. 

151 

49.4802 

194.828 

151 

49.8729 

197.933 

16 

50.2656 

201.062 

161 

50.6583 

204.216 

161 

51.0510 

207.395 

16| 

51.4437 

210.598 

161 

51.8364 

213.825 

16| 

52.2291 

217.077 

161 

52.6218 

220.354 

161 

53.0145 

223.655 

17 

53.4072 

226.981 

171 

53.7999 

230.331 

171 

54.1926 

233.706 

17| 

54.5853 

237.105 

171 

54.9780 

240.529 

17| 

55.3707 

243.977 

171 

55.7634 

247.450 

171 

56.1561 

250.948 

18 

56.5488 

254.470 

181 

56.9415 

258.016 

181 

57.3342 

261.587 

181 

57.7269 

265.183 

181 

58.1196 

268.803 

18| 

58.5123 

272.448 

181 

58.9050 

276.117 

181 

59.2977 

279.811 

19 

59.6904 

283.529 

191 

60.0831 

287.272 

191 

60.4758 

291.040 

19| 

60.8685 

294.832 

191 

61.2612 

298.648 

191 

61.6539 

302.489 

191 

62.0466 

306.355 

191 

62.4393 

310.245 

20 

62.8320 

314.160 

201 

63.2247 

318.099 

201 

63.6174 

322.063 

201 

64.0101 

326.051 

201 

64.4028 

330.064 

20| 

64.7955 

334.102 

201 

65.1882 

338.164 

201 

65.5809 

342.250 

21 

65.9736 

346.361 

211 

66.3663 

350.497 

211 

66.7590 

354.657 



















122 


AREAS AND CIRCUMFERENCE OF CIRCLES. 


TABLE—CONTINUED. 


Diam. 

Circum. 

Area. 

211 

67.1517 

358.842 

21| 

67.5444 

363.051 

21| 

67.9371 

367.285 

21| 

68.3298 

371.543 

211 

68.7225 

375.826 

22 

69.1152 

380.134 

22| 

69.5079 

384.466 

22\ 

69.9006 

388.822 

221 

70.2933 

393.203 

22\ 

70.6860 

397.609 

22| 

71.0787 

402.038 

22| 

71.4714 

406.494 

221 

71.8641 

410.973 

23 

72.2568 

415.477 

231 

72.6495 

420.004 

231 

73.0422 

424.558 

231 

73.4349 

429.135 

231 

73.8276 

433.737 

23! 

74.2203 

438.364 

23! 

74.6130 

443.015 

231 

75.0057 

| 447.690 

24 

75.3984 

452.390 

241 

75.7911 

457.115 

241 

76.1838 

461.864 

241 

76.5765 

466.638 

241 

76.9692 

471.436 

24| 

77.3619 

476.259 

24! 

77.7546 

481.107 

241 

78.1473 

485.979 

25 

78.5400 

490.875 

251 

78.9327 

495.796 

251 

79.3254 

500.742 

251 

79.7181 

505.712 

251 

80.1108 

510.706 

25! 

80.5035 

515.726 

25! 

80.8962 

520.769 

251 

81.2889 

525.838 

26 

81.6816 

530.930 

261 

82.0743 

536.048 

261 

82.4670 

541.190 

261 

82.8597 

546.356 

261 

83.2524 

551.547 

26| 

83.6451 

556.763 

26! 

84.0378 

562.003 

261 

84.4305 

567.267 


Diam. 

Circum. ■* 

Area. 

21 

84.8232 

572.557 

27| 

85.2159 

577.870 

271 

85.6086 

583.209 

271 

86.0013 

588.571 

271 

86.3940 

593.959 

27! 

86.7867 

599.371 

27! 

87.1794 

604.807 

271 

87.5721 

610.268 

28 

87.9648 

615.754 

281 

88.3575 

621.264 

281 

88.7502 

626.798 

281 

89.1429 

632.357 

281 

89.5356 

637.941 

28! 

89.9283 

643.549 

28! 

90.3210 

649.182 

281 

90.7137 

654.840 

29 

91.1064 

660.521 

291 

91.4991 

666.228 

291 

91.8918 

671.959 

291 

92.2845 

677.714 

291 

92.6772 

683.494 

29! 

93.0699 j 

689.299 

29! 

93.4626 ] 

| 695.128 

291 

93.8553 

! 700.982 

30 

94.2480 

| 706.860 

301 

94.6407 

| 712.763 

301 

95.0334 

| 718.690 

301 

95.4261 

| 724.642 

301 

95.8188 

| 730.618 

301 

96.2115 

! 736.619 

30! 

96.6042 

! 742.645 

301 

96.9969 

1 748.695 

31 

97.3896 

754.769 

311 

97.7823 . 

760.869 

311 

98.1750 

766.992 

311 

98.5677 

l 773.140 

311 

98.9604 

779.313 

31! 

99.3531 

785.510 

31! 

99.7458 

791.732 

311 

100.1385 

I 797.979 

32 

100.5312 

| 804.250 

321 

100.9239 

i 810.545 

321 

101.3166 

! 816.865 

32! 

101.7093 

| 823.210 

321 

102.1020 

| 829.579 
























AREAS AND CIRCUMFERENCE OF CIRCLES. 


123 


TABLE—CONTINUED. 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

32| 

102.4947 

835.972 

381 

120.166 

1,149.089 

321 

102.8874 

842.391 

38| 

120.559 

1,156.612 

321 

103.280 

848.833 

38! 

120.952 

1,164.159 

33 

103.673 

855.301 

381 

121.344 

1,171.731 

33| 

104.065 

861.792 

38| 

121.737 

1,179.327 

33| 

104.458 

868.309 

381 

122.130 

1,186.948 

33| 

104.851 

874.850 

39 

122.522 

1,194.593 

33! 

105.244 

881.415 

39| 

122.915 

1,202.263 

33| 

105.636 

888.005 

391 

123.308 

1,209.958 

332 

106.029 

894.620 

39| 

123.700 

1,217.677 

331 

106.422 

901.259 

39! 

124.093 

1,225.420 

34 

106.814 

907.922 

391 

124.486 

1,233.188 

341 

107.207 

914.611 

391 

124.879 

1,240.981 

34! 

107.600 

921.323 

391 

125.271 

1,248.798 

34| 

107.992 

928.061 

40 

125.664 

1,256.640 

341 

108.385 

934.822 

40| 

126.057 

1,264.510 

34| 

108.778 

941.609 

401 

126.449 

1,272.400 

341 

109.171 

948.420 

401 

126.842 

1,280.310 

341 

109.563 

955.255 

40! 

127.235 

1,288.250 

35 

109.956 

962.115 

401 

127.627 

1,296.220 

351 

110.349 

969.000 

401 

128.020 

1,304.210 

351 

110.741 

975.909 

401 

128.413 

1,312.220 

351 

111.134 

982.842 

41 

128.806 

1,320.260 

351 

111.527 

989.800 

41! 

129.198 

1,328.320 

35| 

111.919 

996.783 

411 

129.591 

1,336.410 

351 

112.312 

1.003.790 

41| 

129.984 

1,344.520 

. 351 

112.705 

1,010.822 

41! 

130.376 

1,352.660 

36 

113.098 

1.017.878 

411 

130.769 

1,360.820 

361 

113.490 

1,024.960 

411 

131.162 

1,369.000 

361 

113.883 

1,032.065 

411 

131.554 

1,377.210 

36| 

114.276 

1,039.195 

42 

131.947 

1,385.450 

36! 

114.668 

1,046.349 

42! 

132.340 

1,393.700 

36| 

115.061 

1,053.528 

421 

132.733 

1,401.990 

361 

115.454 

1,060.732 

42g 

133.125 

1,410.300 

361 

115.846 

1,067.960 

42! 

133.518 

1,418.630 

37 

116.239 

1,075.213 

421 

133.911 

1,426.990 

371 

116.632 

1,082.490 

421 

134.303 

1,435.370 

371 

117.025 

1,089.792 

421 

134.696 

1,443.770 

371 

117.417 

1,097.118 

43 

135.089 

1,452.200 

37! 

117.810 

1,104.469 

43! 

135.481 

1,460.660 

37p 

118.203 

1,111.844 

431 

135.874 

1,469.140 

371 

118.595 

1,119.244 

431 

136.267 

1,477.640 

371 

118.988 

1,126.669 

43! 

136.660 

1,486.170 

38 

119.381 

1,134.118 

431 

137.052 

1,494.730 

381 

119.773 

1,141.591 

431 

137.445 

1,503.300 






















124 


AREAS AND CIRCUMFERENCE OF CIRCLES. 


TABLE—CONTINUED. 


Dlam. 

Circum. 

Area. 

431 

137.838 

1,511.91 

44 

138.230 

1,520.53 

441 

138.623 

1,529.19 

441 

139.016 

1,537.86 

44g 

139.408 

1,546.56 

44| 

139.801 

1,555.29 

44| 

140.194 j 

1,564.04 

441 

140.587 ! 

1,572.81 

441 

140.979 

1,581.61 

45 

141.372 | 

1,590.43 

451 

141.765 

1,599.28 

451 

142.157 

1,608.16 

45| 

142.550 

1,617.05 

45! 

142.943 

1,625.97 

45| 

143.335 

1,634.92 

451 

143.728 

1,643.89 

451 

144.121 

1,652.89 

46 

144.514 

1,661.91 

461 

144.906 

1,670.95 

461 

145.299 

1,680.02 

461 

145.692 

1,689.11 

46! 

146.084 

1,698.23 

46| 

146.477 

1,707.37 

461 

146.870 

1,716.54 

461 

147.262 

1,725.73 

47 

147.655 

1,734.95 

471 

148.048 

1,744.19 

471 

148.441 

1,753.45 

471 

148.833 

1,762.74 

47! 

149.226 

1,772.06 

47| 

149.619 

1,781.40 

471 

150.011 

1,790.76 

471 

150.404 

1,800.15 

48 

150.797 

1 1,809.56 

481 

151.189 

1,819.00 

481 

151.582 

1,828.46 

481 

151.975 

1,837.95 

48! 

152.368 

1,847.46 

48| 

152.760 

1,856.99 

481 

153.153 

1,866.55 

481 

153.546 

1,876.14 

49 

153.938 

| 1,885.75 

491 

154.331 

| 1,895.38 

491 

154.724 

| 1,905.04 

491 

155.116 

| 1,914.72 


Diam. 

Circum. 

Area. 

49! 

155.509 

1,924.43 

49! 

155.902 

1,934.16 

491 

156.295 

1,943.91 

491 

156.687 

1,953.69 

50 

157.080 

1,963.50 

50! 

157.473 

1,973.33 

501 

157.865 

1,983.18 

50| 

158.258 

1,993.06 

50! 

158.651 

2,002.97 

50| 

159.043 

2,012.89 

501 

159.436 

2,022.85 

501 

159.829 

2,032.82 

51 

160.222 

2,042.83 

51! 

160.614 

2,052.85 

511 

161.007 

2,062.90 

511 

161.400 

2,072.98 

51! 

161.792 

2,083.08 

51| 

162.185 

2,093.20 

511 

162.578 

2,103.35 

511 

162.970 

2,113.52 

52 

163.363 

2,123.72 

52| 

163.756 

2,133.94 

521 

164.149 

2,144.19 

521 

164.541 

2,154.46 

52! 

164.934 

2,164.76 

52| 

165.327 

2,175.08 

521 

165.719 

2,185.42 

521 

166.112 

2,195.79 

53 

166.505 

2,206.19 

53! 

166.897 

2,216.61 

531 

167.290 

2,227.05 

531 

167.683 

2,237.52 

53! 

168.076 

2,248.01 

53! 

168.468 

2,258.53 

531 

168.861 

2,269.07 

531 

169.254 

2,279.64 

54 

169.646 

2,290.23 

54! 

170.039 

2,300.84 

541 

170.432 

2,311.48 

541 

170.824 

2,322.15 

54! 

171.217 

2,332.83 

54! 

171.610 

2,343.55 

541 

172.003 

2,354.29 

541 

172.395 

2,365.05 

55 

| 172.788 

2,375.83 
























AREAS AND CIRCUMFERENCE OF CIRCLES. 


125 


TABLE—CONTINUED. 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

55| 

173.181 i 

2,386.65 

601 

190.852 

2,898.57 

55J 

173.573 I 

2,397.48 

601 

191.245 

2,910.51 

55| 

173.966 ! 

2,408.34 

61 

191.638 

2,922.47 

55i 

174.359 

2,419.23 

611 

192.030 

2,934.46 

55| 

174.751 

2,430.14 

611 

192.423 

2,946.48 

551 

175.144 

2,441.07 

61| 

192.816 

2,958.52 

551 

175.537 

2,452.03 

611 

193.208 

2,970.58 

56 

175.930 

2,463.01 

61§ 

193.601 

2,982.67 

561 

176.322 

2,474.02 

611 

193.994 

2,994.78 

561 

176.715 

2,485.05 

611 

194.386 

3,006.92 

561 

177.108 

2,496.11 

62 

194.779 

3,019.08 

561 

177.500 

2,507.19 

621 

195.172 

3,031.26 

56| 

177.893 

2,518.30 

621 

195.565 

3,043.47 

561 

178.286 

2,529.43 

621 

195.957 

3,055.71 

561 

178.678 

2,540.58 

621 

196.350 

3,067.97 

57 

179.071 

2,551.76 

62| 

196.743 | 

3,080.25 

571 

179.464 

2.562.97 

621 

197.135 I 

1 3,092.56 

571 

179.857 

2,574.20 

621 

197.528 

3,104.89 

571 

180.249 

2,585.45 

63 

197.921 

3,117.25 

571 

180.642 

2,596.73 

631 

198.313 

3,129.64 

571 

181.035 

2,608.03 

631 

198.706 

3,142.04 

571 

181.427 

2,619.36 

63| 

199.099 

3,154.47 

571 

181.820 

2,630.71 

631 

199.492 

3,166.93 

58 

182.213 

2,642.09 

631 

199.884 

3,179.41 

581 

182.605 

2,653.49 

631 

200.277 

3,191.91 

581 

182.998 

2,664.91 

631 

200.670 

3,204.44 

581 

183.391 

2,676.36 

64 

201.062 

3,217.00 

581 

183.784 

2,687.84 

641 

201.455 

3,229.58 

581 

184.176 

2,699.33 

641 

201.848 

3,242.18 

581 

184.569 

2,710.86 

64| 

202.240 

3,254.81 

581 

184.962 

2,722.41 

641 

202.633 

3,267.46 

59 

185.354 

2,733.98 

64| 

203.026 

3,280.14 

591 

185.747 

2,745.57 

641 

203.419 

3,292.84 

591 

186.140 

2,757.20 

641 

203.811 

3,305.56 

591 

186.532 

2,768.84 

65 

204.204 

3,318.31 

591 

186.925 

: 2,780.51 

651 

204.597 

3,331.09 

591 

187.318 

2,792.21 

651 

204.989 

3,343.89“ 

591 

187.711 

2,803.93 

651 

205.382 

3,356.71 

591 

188.103 

2,815.67 

651 

205.775 

3,369.56 

60 

188.496 

2,827.44 

65| 

206.167 

3,382.44 

601 

188.889 

2,839.23 

651 

206.560 

3,395.33 

601 

189.281 

2,851.05 

651 

206.953 

3,408.26 

601 

189.674 

| 2,862.89 

66 

207.346 

3,421.20 

601 

190.067 

2,874.76 

661 

207.738 

3,434.17 

60| 

190.459 

| 2,886.65 

661 

208.131 

1 3,447.17 




























126 


AREAS AND CIRCUMFERENCE OF CIRCLES. 


TABLE—CONTINUED. 


Diam. 

Circum. 

Area. 

66| 

208.524 

1 3,460.19 

661 

208.916 

3,473.24 

66| 

209.309 

3,486.30 

66| 

209.702 

3,499.40 

661 

210.094 

3,512.52 

67 

210.487 

3,525.66 

671 

210.880 

3,538.83 

671 

211.273 

3,552.02 

67| 

211.665 

3,565.24 

671 

212.058 

3,578.48 

671 

212.451 

3,591.74 

671 

212.843 

3,605.04 

671 

213.236 

3,618.35 

68 

213.629 

3,631.69 

681 

214.021 

3,645.05 

681 

214.414 

3,658.44 

68| 

214.807 

3,671.86 

681 

215.200 

3,685.29 

68| 

215.592 

3,698.76 

681 

215.985 

3,712.24 

681 

216.378 j 

I 3,725.75 

69 

216.770 i 

3,739.29 

691 

217.163 

j 3,752.85 

691 

217.556 

3,766.43 

691 

217.948 

3,780.04 

691 

218.341 

3,793.68 

69| 

218.734 

3,807.34 

691 

219.127 

3,821.02 

691 

219.519 

3,834.73 

70 

219.912 

3,848.46 

701 

220.305 

3,862.22 

701 

220.697 

3,876.00 

701 

221.090 

3,889.80 

701 

221.483 

3,903.63 

70| 

221.875 

3,917.49 

701 

222.268 

3,931.37 

701 

222.661 

3,945.27 

71 

223.054 

3,959.20 

711 

223.446 

3,973.15 

711 

223.839 

3,987.13 

711 

224.232 

4,001.13 

711 

224.624 

4,015.16 

71§ 

225.017 ! 

4,029.21 

711 

225.410 

4,043.29 

711 

225.802 

4,057.39 


Diam. 

Circum. 

Area. 

72 

226.195 

4,071.51 

721 

226.588 

4,085.66 

72J 

226.981 

4,099.84 

72| 

227.373 

4,114.04 

72£ 

227.766 

4,128.26 

72| 

228.159 

4,142.51 

721 

228.551 

4,156.78 

72| 

228.944 

4,171.08 

73 

229.337 

4,185.40 

731 

229.729 

4,199.74 

731 

230.122 

4,214.11 

731 

230.515 

4,228.51 

731 

230.908 

4,242.93 

75| 

231.300 

4,257.37 

731 

231.693 

4,271.84 

731 

232.086 

4,286.33 

74 

232.478 

4,300.85 

74| 

232.871 

4,315.39 

741 

233.264 

4,329.96 

74§ 

233.656 

4,344.55 

741 

234.049 

4,359.17 

74§ 

234.442 

4,373.81 

741 

234.835 

! 4,388.47 

741 

235.227 

4,403.16 

75 

235.620 

4,417.87 

751 

236.013 

4,432.61 

751 

236.405 

4,447.38 

751 

236.798 

4,462.16 

751 

237.191 

4,476.98 

75| 

237.583 

4,491.81 

751 

237.976 

4,506.67 

751 

238.369 

4,521.56 

76 

238.762 

4,536.47 

761 

239.154 

4,551.41 

761 

239.547 

4,566.36 

761 

239.940 

4,581.35 

761 

240.332 

4,596.36 

76| 

240.725 

4,611.39 

761 

241.118 

4,626.45 

761 

241.510 

4,641.53 

77 

241.903 

4,656.64 

771 

242.296 

4,671.77 

771 

242.689 

4,686.92 

771 

243.081 

| 4,702.10 

771 

243.474 

| 4,717.31 



























Diam. 

77| 

771 

771 

78 

78i 

781 

781 

78 h 

78| 

781 

781 

79 

791 

79J 

79| 

791 

79§ 

791 

791 

80 

801 

801 

80| 

801 

80| 

801 

801 

81 

811 

811 

811 

811 

81| 

811 

711 

82 

821 

821 

821 

821 

82| 

821 

821 

83 

831 


AND CIRCUMFERENCE OF CIRCLES. 


127 


TABLE—CONTINUED. 


Circum. 

Area. 

Diam. 

Circum. 

Area. 

243.867 

4,732.54 

83| 

261.538 

| 5,443.26 

244.259 

4,747.79 

83| 

261.931 

| 5,459.62 

244.652 

4,763.07 

83i 

262.324 

! 5,476.01 

245.045 

4,778.37 

831 

262.716 

5,492.41 

245.437 

4,793.70 

83| 

263.109 

5,508.84 

245.830 

4,809.05 

831 

263.502 

5,525.30 

246.223 

4,824.43 

84 

263.894 

5,541.78 

246.616 

4,839.83 

84| 

264.287 

5,558.29 

247.008 

4,855.26 

84| 

264.680 

5,574.82 

247.401 

4,870.71 

84| 

265.072 

5,591.37 

247.794 

4,886.18 

84J 

265.465 

5,607.95 

248.186 

4,901.68 

841 

265.858 

5,624.56 

248.579 

4,917.21 

84| 

266.251 

5,641.18 

248.972 

4,932.75 

841 

266.643 

5,657.84 

249.364 

4,948.33 

85 

267.036 

5,674.51 

249.757 

4,963.92 

85i 

267.429 

5,691.22 

250.150 

4,979.55 

85| 

267.821 

5,707.94 

250.543 

4,995.19 

85| 

268.214 

5,724.69 

250.935 

5,010.86 

85! 

268.607 

5,741.47 

251.328 

5,026.56 

851 

268.999 

5,758.27 

251.721 

5,042.28 

851 

269.392 

5,775.10 

252.113 

5,058.03 

851 

269.785 

5,791.94 

252.506 

5,073.79 

86 

270.178 

5,808.82 

252.899 

5,089.59 

861 

270.570 

5,825.72 

253.291 

5,105.41 

86| 

270.963 

5,842.64 

253.684 

5,121.25 

861 

271.356 

5,859.59 

254.077 

5,137.12 

861 

271.748 

5,876.56 

254.470 

5,153.01 

861 

272.141 

5,893.55 

254.862 

5,168.93 

861 

227.534 

5,910.58 

255.255 

5,184.87 

861 

272.926 

5,927.62 

255.648 

5,200.83 

87 

273.319 | 

5,944.69 

256.040 

5,216.82 

871 

273.712 

5,961.79 

256.433 

5,232.84 

87! 

274.105 

5.978.91 

256.826 

5,248.88 

871 

274.497 

5,996.05 

257.218 

5,264.94 

871 

274.890 

6,013.22 

257.611 

5,281.03 

871 

275.283 

6,030.41 

258.004 

5,297.14 

871 

275.675 

6,047.63 

258.397 

5,313.28 

871 

276.068 

6,064.87 

258.789 

5,329.44 

88 

276.461 

6,082.14 

259.182 

5,345.63 

881 

276.863 

6,099.43 

259.575 

5,361.84 

88! 

277.246 

6,116.74 

259.967 

5,378.08 

881 

277.629 

6,134.08 

260.360 

5,394.34 

881 

278.032 

6,151.45 

260.753 

5,410.62 

881 

278.424 

6,168.84 

261.145 

5,426.93 

881 

278.817 

6,186.25 





















128 


AREAS AND CIRCUMFERENCE OF CIRCLES. 


TABLE—CONTINUED. 


Diam. 

Oircum. 

Area. 

881 

279.210 

6,203.69 

89 

279.602 

6,221.15 

891 

279.995 

6,238.64 

891 

280.388 

6,256.15 

89| 

280.780 

6,273.69 

89i 

281.173 

6,291.25 

89| 

281.566 

6,308.84 

891 

281.959 

6,326.45 

891 

282.351 

6,344.08 

90 

282.744 

6,361.74 

901 

283.137 

6,379.42 

901 

283.529 

6,397.13 

901 

283.922 

6,414.86 

901 

284.315 

6,432.62 

90| 

284.707 

6,450.40 

901 

285.100 

6,468.21 

901 

285.493 

6,486.04 

91 

285.886 

6,503.90 

911 

286,278 

6,521.78 

911 

286.671 

6,539.68 

911 

287.064 

6,557.61 

911 

287.456 

6,575.56 

911 

287.849 

6,593.54 

911 

288.242 

6,611.55 

911 

288.634 

6,629.57 

92 

289.027 

6,647.63 

921 

289.420 

6,665.70 

921 

289.813 

6,683.80 

921 

290.205 

6,701.93 

921 

290.598 

6,720.08 

92| 

290.991 

6,738.25 

921 

291.383 

6,756.45 

921 

291.776 

6,774.68 

93 

292.169 

6,792.92 

931 

292.562 

6,811.20 

931 

292.954 

6,829.49 

931 

293.347 

6,847.82 

931 

293.740 

6,866.16 

93| 

294.132 

6,884.53 

931 

294.525 

6,902.93 

931 

294.918 

6,921.35 

94 

295.310 

6,939.79 

941 

295.703 

6,958.26 

941 

296.096 

6,976.76 

941 

296.488 

6,995.28 


Diam. 

Circum. 

Area. 

94 h 

296.881 

7,013.82 

94| 

297.274 

7,032.39 

941 

297.667 

7,050.98 

941 

298.059 

7,069.59 

95 

298.452 

7,088.24 

951 

298.845 

7,106.90 

951 

299.237 

7,125.59 

95| 

299.630 

7,144.31 

951 

300.023 

7,163.04 

95| 

300.415 

7,181.81 

951 

300.808 

7,200.60 

951 

301.201 

7,219.41 

96 

301.594 

7,238.25 

961 

301.986 

7,257.11 

961 

302.379 

7,275.99 

96| 

302.772 

7,294.91 

961 

303.164 

7,313.84 

96| 

303.557 I 

7,332.80 

961 

303.950 I 

7,351.79 

961 

304.342 

7,370.79 

97 

304.735 ! 

7,389.83 

971 

305.128 

7,408.89 

971 

305.521 

7,427.97 

97! 

305.913 

7,447.08 

971 

306.306 

7,466.21 

97| 

306.699 

7,485.37 

971 

307.091 

7,504.55 

971 

307.484 

7,523.75 

98 

307.877 

7,542.98 

981 

308.270 

7,562.24 

981 

308.662 

7,581.52 

98! 

309.055 

7,600.82 

981 

309.448 

7,620.15 

98| 

309.840 

7,639.50 

981 

310.233 

7,658.88 

981 

310.626 

7,678.28 

99 

311.018 

7,697.71 

991 

311.411 

7,717.16 

991 

311.804 

7,736.63 

99! 

312.196 

7,756.13 

991 

312.589 

7,775.66 

991 

312.982 

I 7,795.21 

991 

313.375 

7,814.78 

991 

313.767 

7,834.38 

100 

314.160 

7,854.00 




























MEASURES. 


129 


MEASURES. 


MEASURES OF EXTENSION. 

LINEAR MEASURE. 

Abbreviation 


12 inches 

= 1 foot 

ft. 

3 feet 

= 1 yard 

yd. 

5.5 yards 

= 1 rod 

rd. 

40 rods 

= 1 furlong 

fur. 

8 furlongs 

= 1 mile 

mi. 




SQUARE 

MEASURE. 


144 

square 

inches = 

1 square foot 

sq. ft, 

9 

square 

feet = 

1 square yard 

sq. yd 

30J square 

yards = 

1 square rod 

sq. rd. 

160 

square 

rods = 

1 square acre 

A. 

640 

acres 


1 square mile 

sq. mi. 




CUBIC MEASURE. 


,728 

cubic 

inches 

= 1 cubic foot 

cu. ft. 

27 

cubic 

feet 

= 1 cubic yard 

cu. yd. 

128 

cubic 

feet 

= 1 cord 

cd. 

241 

cubic 

feet 

= 1 perch 

P. 



130 


MEASURES. 


MEASURES OF WEIGHT. 

AVOIRDUPOIS WEIGHT. 

16 ounces = 1 pound lb. 

100 pounds = 1 hundred we’ght cwt. 

20 cwt., or 2,000 lbs. = 1 ton T. 

LONG TON TABLE. 

16 ounces = 1 pound lb. 

112 pounds = 1 hundred we’ght cwt. 

20 cwt., or 2,240 lbs. = 1 ton T. 

TROY WEIGHT. 

24 grains = 1 pennyweight pwt. 

20 pennyweights = 1 ounce os. 

12 ounces = 1 pound lb. 

Note —Troy weight used by jewelers. 


MEASURES OF CAPACITY. 

LIQUID MEASURE. 

4 gills = 1 pint pt. 

2 pints = 1 quart qt. 

4 quarts = lgallon gal. 

31J gallons = 1 barrel bbl. 

2 barrels = 1 hogshead hhd. 

DRY MEASURE. 

2 pints = 1 quart qt. 

8 quarts = 1 peck pk. 

4 pecks = 1 bushel bu. 




MEASURES. 


131 


MEASURE OF TIME. 


60 

seconds 

= 

1 minute 

min. 

60 

minutes 

= 

1 hour 

hr. 

24 

hours 

= 

1 day 

da. 

7 

days 

= 

1 week 

wk. 

365 

days 

= 

1 common year 

yr. 

366 

days 

= 

1 leap year 

yr. 

100 

years 

— 

1 century 


Note—T hirty 

days is generally considered 

a month 


MEASURES OF ANGLES OR ARCS. 


60 

seconds 

= 1 minute 

// 

60 

minutes 

= 1 degree 

O 

90 

degrees 

= 1 right angle 

L. 

360 

degrees 

= 1 circle 

cir. 


MEASURES OF MONEY. 


10 

mills 

= 1 cent 

ct. 

10 

cents 

= 1 dime 

d. 

10 

dimes 

= 1 dollars 

<* 

V 

10 

dollars 

= 1 eagle 

E. 


MISCELLANEOUS TABLE. 


12 things are 1 dozen. 

12 dozen are 1 gross. 

12 gross are 1 great gross. 

2 things are 1 pair. 

20 things are 1 score. 

1 league is 3 miles. 

1 fathom is 6 feet. 


1 meter is 39.37 inches. 

1 hand is 4 inches. 

1 palm is 3 inches. 

1 span is 9 inches. 

24 sheets are 1 quire. 

20 quires are 1 ream. 

1 bu. contains 2,150.4 cu. in. 


1 United States gallon contains 231 cu. in. 

1 United States gallon of water weighs 8.355 pounds, 
nearly. 

1 British imperial gallon of water weighs 10 pounds. 

1 cubic foot of water contains 7.481 United States 
standard gallons. 






132 


DECIMALS OF A FOOT. 


DECIMALS OF A FOOT FOR EACH 3 \nd OF 
AN INCH. 


Inch 

0" 

1" 

2" 

3" 

4" 

5" 

6" 

7" 

8" 

9" 

10" 

11" 

0 

0 

.0833 

.1667 

.2500 

.3333 

.4167 

5000 

.5833 

.6667 

.7500 

.8333 

.9167 

ft 

.0026 

.0859 

.1693 

.2526 

.3359 

.4193 

.5026 

.5859 

.6693 

.7526 

.8359 

.9193 

ft 

.0052 

.0885 

.1719 

.2552 

.3385 

.4219 

.5052 

.5885 

.6719 

.7552 

.8385 

.9219 

ft 

.0078 

.0911 

.1745 

.2578 

.3411 

.4245 

.5078 

.5911 

.6745 

.7578 

.8411 

.9245 

1 

.0104 

.0937 

.1771 

.2604 

.3437 

.4271 

.5104 

.5937 

.6771 

.7604 

.8437 

.9271 

ft 

.0130 

.0964 

.1797 

.2630 

.3464 

.4297 

.5130 

.5964 

.6797 

.7630 

.8464 

.9297 

ft 

.0156 

.0990 

.1823 

.2656 

.3490 

.4323 

.5156 

.5990 

.6823 

.7656 

.8490 

.9323 

ft 

.0182 

.1016 

.1849 

.2682 

.3516 

.4349 

.5182 

.6016 

.6849 

.7682 

.8516 

.9349 

J 

.0208 

.1042 

.1875 

.2708 

.3542 

.4375 

.5208 

.6042 

.6875 

.7708 

.8542 

.9375 

ft 

.0234 

.1068 

.1901 

.2734 

.3568 

.4401 

.5234 

.6068 

.6901 

.7734 

.8568 

.9401 

ft 

.0260 

.1094 

.1927 

.2760 

.3594 

.4427 

.5260 

.6094 

.6927 

.7760 

.8594 

.9427 

u 

.0286 

.1120 

.1953 

.2786 

.3620 

.4453 

.5286 

.6120 

.6953 

.7786 

.8620 

.9453 

1 

.0312 

.1146 

.1979 

.2812 

.3646 

.4479 

.5312 

.6146 

.6979 

.7812 

.8646 

.9479 

H 

.0339 

.1172 

.2005 

.2839 

.3672 

.4505 

.5339 

.6172 

.7005 

.7839 

.8672 

.9505 

ft 

.0365 

.1198 

.2031 

.2865 

.3698 

.4531 

.5365 

.6198 

.7031 

.7865 

.8698 

.9531 

l§ 

.0391 

.1224 

.2057 

.2891 

.3724 

.4557 

.5391 

.6224 

.7057 

.7891 

.8724 

.9557 

i 

.0417 

.1250 

.2083 

.2917 

.3750 

.4583 

.5417 

.6250 

.7083 

.7917 

.8750 

.9583 

H 

.0443 

.1276 

.2109 

.2943 

.3776 

.4609 

.5443 

.6276 

.7109 

.7943 

.8776 

.9609 

■ft 

.0469 

.1302 

.2135 

.2969 

.3802 

.4635 

.5469 

.6302 

.7135 

.7969 

.8802 

.9635 

IS 

.0495 

.1328 

.2161 

.2995 

.3828 

.4661 

.5495 

.6328 

.7161 

.7995 

.8828 

.9661 

i 

.0521 

.1354 

.2188 

.3021 

.3854 

.4688 

.5521 

.6354 

.7188 

.8021 

.8854 

.9688 

II 

.0547 

.1380 

.2214 

.3047 

.3880 

.4714 

.5547 

.6380 

.7214 

.8047 

.8880 

.9714 

tt 

.0573 

.1406 

.2240 

.3073 

.3906 

.4740 

.5573 

.6406 

.7240 

.8073 

.8906 

.9740 

§1 

.0599 

.1432 

.2266 

.3099 

.3932 

.4766 

.5599 

.6432 

.7266 

.8099 

.8932 

.9766 

1 

.0625 

.1458 

.2292 

.3125 

.3958 

.4792 

.5625 

.6458 

.7292 

.8125 

.8958 

.9792 

II 

.0651 

.1484 

.2318 

.3151 

.3984 

.4818 

.5651 

.6484 

.7318 

.8151 

.8984 

.9818 

» 

.0677 

.1510 

.2344 

.3177 

.4010 

.4844 

.5677 

.6510 

.7344 

.8177 

.9010 

.9844 

§1 

.0703 

.1536 

.2370 

.3203 

.4036 

.4870 

.5703 

.6536 

.7370 

.8203 

.9036 

.9870 

i 

.0729 

.1562 

.2396 

.3229 

.4062 

.4896 

.5729 

.6562 

.7396 

.8229 

.9062 

.9896 

II 

.0755 

.1589 

.2422 

.3255 

.4089 

.4922 

.5755 

.6589 

.7422 

.8255 

.9089 

.9922 

II 

.0781 

.1615 

.2448 

.3281 

.4115 

.4948 

.5781 

.6615 

.7448 

.8281 

.9115 

.9948 

II 

l 

.0807 

.1641 

.2474 

.3307 

.4141 

.4974 

.5807 

.6641 

.7474 

.8307 

.9141 

.9974 

1.0000 






























NUMBER OF GALLONS IN ROUND CISTERNS AND TANKS. 


NUMBER OF GALLONS 


133 


Depth in 

ID 

CD 

tH 

CO 

05 

O 

rH 

cq 

CO 

rH 

LD 

CD 

ch 

GO 

05 

O 


jseec. 






rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

tH 

rH 

cq 



00 

O 

rH 

oq 

CO 

rH 

LD 

CD 

tH 

OO 

05 

O 

rH 

cq 

CO 

rH 


25 

LD 

CO 

O 

tH 

rH 

rH 

00 

LD 

cq 

05 

CD 

rH 

rH 

OO 

LD 

cq 


CO 

o 

tH 

CO 

O 

tH 

** 

CO 

O 

•> 

tH 

•s 

CO 

O 

tH 

rH 

•v 

o 

tH 

r\ 

rH 



00 

oq 

LD 

05 

CO 

CD 

o 

rH 

tH 

rH 

LD 

OO 

cq 

CD 

05 

CO 



T—1 

oq 

oq 

oq 

CO 

CO 

rH 


rH 

LD 

LD 

LD 

CD 

CD 

CD 

tH 



00 

oq 

o 

o 

rH 

00 

cq 

CD 

O 

rH 

00 

cq 

O 

O 

rH 

00 



rH 

o 

00 

tH 

LD 

CO 

cq 

O 

05 

tH 

LD 

rH 

cq 

rH 

05 

tH 


rp 

<M 

05^ 

co 

CD 

© 

rH 

oo 

cq 

CD 

O 

CO 

tH 

rH 

LD 

05 

cq 

CD 


CD 

o 

co" 

tH 

© 

CO 

tH 

o' 

cd' 

ih' 

O 

rH 

tH 

o' 

rH 

tH 



rH 

oq 

oq 

oq 

CO 

CO 

co 

rH 

rH 

rH 

LD 

LD 

LD 

CD 

CD 

CD 



ID 

05 

oq 

LD 

00 

rH 

rH 

IH 

O 

CO 

CD 

05 

cq 

LD 

00 

rH 


22 

rH 

LD 

o 

rH 

00 

CO 

tH 

rH 

CD 

O 

rH 

OO 

CO 

tH 

tH 

CD 


N 

© 

05 

tH 

LD 

rH 

cq 

tH 

05 

OO 

CD 

rH 

CO 

rH 

O 

00 


rH 

tH 

05 

Oq 

ld 

00' 

rH 

rH 

cd" 

05* 

cq 

LD 

od' 

rH 

rH 

CD 



rH 

tH 

rH 

oq 

cq 

cq 

CO 

CO 

CO 

CO 

rH 

rH 

rH 

LD 

LD 

LD 



O 

O 

O 

o 

o 

o 

o 

o 

o 

o 

O 

O 

O 

O 

O 

O 



ID 

O 

LD 

o 

LD 

o 

LD 

o 

LD 

o 

LD 

O 

LD 

O 

LD 

O 


cq 

tH 

t-H^ 

rH 

00 

rH 

LD 

co 

cq 

LD 

05 

cq 

CD 

05 

co 

CD 

O 


rH 

rH 

CD 

00 

rH 

CO 

LD 

oo' 

o' 

cq' 

ld' 

tH 

05* 

Cq" 

rH 

tH 



rH 

rH 

rH 

rH 

cq 

cq 

cq 

cq 

CO 

CO 

CO 

CO 

CO 

rH 

rH 

rH 



CD 

05 

oq 

LD 

00 

tH 

rH 

tH 

O 

CO 

CD 

05 

cq 

LD 

00 

cq 


00 

rH 

tH 

cq 

oq 

cq 

CO 

CO 

CO 

rH 

rH 

rH 

rH 

LD 

LD 

LD 

CD 


LD 

rH 

CO 

oq 

rH 

© 

05 

00 

tH 

CD 

LD 

rH 

CO 

cq 

rH^ 

O 



05 

rH 

CO 

LD 

tH 

05 

o' 

cq 

rH 

cd" 

od' 

o' 

cq' 

rH 

cd’ 

00 




rH 

rH 

rH 

rH 

rH 

cq 

cq 

cq 

cq 

cq 

CO 

co 

CO 

CO 

CO 



o 

rH 

00 

oq 

CD 

O 

rH 

00 

cq 

CD 

o 

rH 

oo 

cq 

CD 

o 


co 

Cq 


oq 

CO 

CO 

rH 

rH 

rH 

LD 

LD 

CD 

CD 

CD 

tH 

tH 

CO 


LD 

O 

LD 

© 

LD 

© 

LD 

O 

LD 

O 

cq 

O 

LD 

o 

LD 

o 



tH 

05 

© 

oq 

co' 

LD*' 

cd" 

oo' 

05 

tH 

cq' 

rH 

ld' 

tH 

oo' 

o' 





rH 

rH 

tH 

rH 

rH 

rH 

tH 

cq 

cq 

cq 

cq 

cq 

cq 

CO 



00 

00 

OO 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

GO 

GO 


to 

05 

CO 

tH 

rH 

LD 

05 

CO 

IH 

rH 

LD 

05 

CO 

IH 

rH 

ID 

05 

Eh 

CD 

© 

CO 

tH 

© 

CO 

tH 

O 

rH 

tH 

O 

rH 

tH 

rH 

rH 

tH 

W 

W 


CD 

00 

05 

© 

cq 

CO 

rH 

cd' 

tH 

cd' 

o' 

rH 

cq" 

rH 

LD' 

CD 





rH 

tH 

rH 

rH 

rH 

rH 

rH 

cq 

cq 

cq 

cq 

oq 

cq 


LD 

OO 

rH 

rH 

tH 

O 

CO 

CD 

05 

cq 

LD 

oo 

rH 

rH 

tH 

o 

rH 

y—j 

CD 

rH 

tH 

oq 

tH 

CO 

GO 

CO 

00 

rH 

05 

rH 

O 

LD 

O 

CD 


tH 

05 

© 

oq 

CO 

LD 

CD 

00 

05 

rH 

cq 

rH 

CD 

tH 

05 

O 



iD* 

CD 

00 

05 

© 

rH 

cq' 

CO 

rH 

cd' 

tH 

oo' 

05 

o' 

rH 

CO 

hH 






rH 

rH 

rH 

rH 

rH 

tH 

tH 

rH 

rH 

cq 

cq 

cq 

W 


O 

oq 

rH 

CD 

OO 

O 

CO 

rH 

CD 

CO 

O 

Cq 

rH 

CD 

00 

o 

CO 

CD 

LD 

rH 

CO 

Cq 

cq 

rH 

O 

05 

OO 

OO 

tH 

CD 

LD 

rH 

rH 

05 

05 

05 

05 

05 

05 

05 

05 

00 

00 

00 

CO 

00 

OO 

CO 

00 

Eh 


rH 

LD 

cd" 

tH 

oo' 

05* 

o' 

rH 

cq' 

co' 

rH 

LD 

cd' 

tH 

oo’ 

05 









rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

tH 

rH 



tH 

rH 

rH 

00 

LD 

cq 

05 

CD 

CO 

O 

tH 

rH 

rH 

GO 

LD 

cq 

< 

cq 

mmmJ 

CO 

OO 

CO 

tH 

cq 

tH 

rH 

CD 

tH 

CD 

o 

LD 

O 

rH 

05 

rH 

cq 

O 

05 

tH 

CD 

rH 

CO 

rH 

O 

OO 

tH 

LD 

rH 

cq 

O 

05 

hH 


rH 

ld‘ 

LD' 

cd' 

tH 

od' 

05 

o' 

rH 

rH 

cq 

cd' 

rH 

LD 

CD 

CD 

Q 









rH 

rH 

tH 

rH 

rH 

tH 

rH 

rH 

rH 



o 

O 

O 

o 

o 

O 

O 

O 

O 

O 

O 

O 

O 

O 

O 

O 



LD 

CD 

tH 

00 

05 

o 

rH 

cq 

CO 

rH 

LD 

CD 

tH 

CO 

05 

O 


rH 
— | 

LD 

cq 

05 

CD 

CO 

rH 

00 

LD 

cq 

05 

CD 

CO 

O 

tH 

rH 

cq 



CO 

rH 

rH 

LD 

CD 

IH 

tH 

00 

05 

05' 

o' 

rH 

cq' 

cq 

CO 

rH 













rH 

rH 

rH 

rH 

rH 

rH 



LD 

O 

LD 

O 

LD 

o 

LD 

O 

LD 

O 

LD 

O 

LD 

O 

LD 

O 




rH 

05 

00 

CD 

LD 

CO 

cq 

O 

05 

tH 

CD 

rH 

CO 

rH 

o 


O 

rH 

05 

LD 

© 

CD 

cq 

00 

rH 

O 

CD 

rH 

tH 

CO 

05 

LD 

rH 

tH 



of 

CO 

rH 

rH' 

LD 

ld" 

CD 

IH 

tH 

00 

00 

05 

05 

o 

rH 

rH 
















rH 

rH 

rH 



o 

LD 

O 

LD 

O 

LD 

O 

LD 

O 

LD 

o 

LD 

O 

LD 

O 

O 


05 

CO 

LD 

CO 

O 

00 

LD 

LD 

O 

OO 

LD 

CO 

O 

CO* 

OO 

rH 

05 


CO 

00 

CO 

00 

cq 

tH 

cq 

tH 

rH 

CD 

tH 

CD 

O 

LD 

O 

rH 



cnT 

oq 

co' 

CO 

rH' 

rH 

ld' 

LD 

cd' 

CD 

tH 

tH 

00 

OO 

05 

oT 



LD 

o 

LD 

o 

LD 

O 

LD 

O 

ID 

O 

LD 

o 

LD 

o 

LD 

o 


GO 

tH 

LD 

oq 

o 

tH 

LD 

cq 

O 

tH 

LD 

cq 

o 

tH 

LD 

cq 

o 


oo 

cq 

CD 

o 

CO 

tH 

tH 

LD 

00 

cq 

CD 

p 

CO 

tH 

rH^ 

LD 



rH 

oq 

oq" 

CO 

co' 

CO 

rH 

rH 

rH 

ld" 

ld' 

cd" 

cd" 

cd" 

tH 

tH 



O 

00 

CD 

rH 

cq 

o 

OO 

CD 

rH 

cq 

O 

00 

CD 

rH 

cq 

O 



rH 

oq 

rH 

O 

05 

00 

CD 

LD 

rH 

CO 

cq 

o 

05 

GO 

tH 

CD 


l> 

rH 

tH 

O 

CO 

LD 

00 

tH 

rH 

tH 

o 

co 

CD 

00 

rH 

rH 

tH 



rH 

rH 

oq 

oq" 

cq 

cq' 

co' 

cd 

CO 

rH' 

rH 

rH' 

rH 

LD 

LD 

ld' 



O 

O 

o 

o 

o 

O 

O 

O 

o 

O 

O 

O 

O 

O 

O 

o 


CD 

CD 

tH 

GO 

05 

o 

rH 

cq 

CO 

rH 

LD 

CD 

tH 

00 

05 

o 

rH 


O 

oq 

rH 

CD 

05 

rH^ 

CO 

LD 

tH 

05 

rH 

CO 

LD 

tH 

p 

cq 



tH 

rH 

rH 

rH 

rH 

cq' 

cq 

cq 

cq 

cq' 

co' 

cd' 

CO 

co' 

rH 

rH' 



LD 

rH 

00 

LD 

cq 

05 

CD 

cq 

05 

CD 

CO 

CD 

tH 

rH 

tH 

00 


ID 

CO 

00 

oq 

tH 

cq 

CD 

rH 

CD 

O 

LD 

o 

LD 

05 

rH 

05 

CO 


tH 

00 

© 

t-H^ 

CO 

rH 

CD 

tH 

05 

O 

cq 

CO 

rH 

CD 

tH 

05 





rH 

rH 

rH 

rH 

rH 

rH 

rH 

cq 

cq 

cq 

Cq" 

cq 

cq 

Cq 

Depth in 

LD 

CD 

tH 

00 

05 

O 

rH 

Cq 

CO 

rH 

LD 

CD 

tH 

00 

05 

O 

Feet. 






rH 

tH 

rH 

rH 

rH 

tH 

tH 

rH 

rH 

rH 

cq 


For tanks that are tapering, measure the diameter four-tenths from the large end. 




























































134 


NUMBER OF RIVETS. 


NUMBER OF RIVETS IN 100 POUNDS. 


Diameter of Rivets. 


M P- 

32 

1 


1 

& 

3 

8 


2 

i 

U 

1 

1 

1 

i 

17500 

15900 

8000 

5100 

3200 

1900 







16000 

13800 

7000 

4500 

2900 

1800 







3 

14400 

12200 

6300 

4100 

2373 

1476 

1103 

642 





g 

13500 

10900 

5700 

3700 

2190 

1371 

1030 

604 





1 

12000 

9800 

5200 

3400 

2034 

1280 

968 

571 

400 

345 



U 

11600 

9000 

4700 

3100 

1898 

1200 

910 

541 

382 

322 

208 

• • • • 

11 

10800 

8300 

4400 

2900 

1780 

1129 

862 

514 

365 

311 

206 

.... 

li 

10000 

7600 

4100 

2700 

1675 

1066 

815 

489 

350 

295 

204 

.... 

li 

9300 

7100 

4000 

2500 

1582 

1010 

776 

462 

335 

284 

201 

• • • • 

li 

8700 

. 

3800 

2300 

1498 

960 

740 

446 

324 

275 

199 

132 

li 

8100 

6300 

3500 

2200 

1424 

914 

707 

428 

311 

266 

192 

128 

H 



3400 

2000 

1356 

872 

672 

411 

302 

257 

185 

124 

2 


5600 

3000 

1900 

1295 

834 

648 

395 

293 

249 

178 

120 

21 





1238 

800 

623 

381 

285 

240 

172 

116 

21 


5000 

2800 

1800 

1187 

768 

599 

367 

277 

233 

167 

112 

2 i 





1139 

738 

577 

354 

269 

226 

162 

108 

21 


4600 

2500 

1700 

1095 

711 

556 

343 

261 

219 

157 

104 

21 





1052 

687 

537 

332 

253 

212 

152 

100 

21 


4200 

2300 

1500 

1017 

662 

519 

321 

245 

206 

148 

96 

21 





982 

636 

503 

311 

237 

201 

144 

92 

3 


3900 

2200 

1400 

949 

611 

487 

302 

230 

196 

140 

88 

31 


3600 

2000 

1300 

890 

581 

459 

285 

218 

186 

132 

85 

31 


3400 

1900 

1200 

837 

548 

433 

270 

208 

177 

126 

82 

3| 


3200 

1800 

1175 

791 

519 

411 

257 

198 

168 

120 

79 

31 







395 

250 

195 

165 

119 


4 


3000 

1700 

1100 

749 

400 

390 

244 

189 

161 

115 

77 

41 



1600 

1050 

700 


372 

233 

180 

155 

no 

75 

41 



1500 

1000 

650 


355 

223 

172 

149 

105 

73 

41 



1475 

925 

625 


339 

214 

166 

143 

101 

71 

5 



1400 

900 

600 


325 

205 

160 

136 

97 

69 

51 



1350 

850 

575 


312 

197 

154 

131 

94 

v £7 

67 

51 



1300 

825 

550 


300 

190 

149 

127 

91 

65 

5f 



1250 

775 

525 


289 

183 

144 

123 

88 

VI V 

63 

6 



1200 

750 

500 


279 

177 

139 

118 

85 

VJO 

61 

61 






171 

135 

114 

82 

59 

61 








165 

131 

110 

79 

57 

61 








160 

127 

JL -L VI 

107 

1 £7 

77 

55 

7 








155 

123 

104 

75 

79 


71 








150 

119 

ivlt 

100 

VO 

51 

71 








146 

116 

lvv 

97 

94 

I o 

71 

49 

47 

71 








142 

113 

110 

1 -L 

69 

67 

8 








138 

i/ 1 * 

92 









•llv 

U 1 

































































REFERENCES. 


135 


REFERENCES 


The radius multiplied by the constant 6.2831 equals the circumfer¬ 
ence. The diameter multiplied by the constant .8862 equals side of 
equal square. The doubling of the diameter of a circle increases its 
area four times. The side of a square multiplied by the constant 1.128 
will equal the diameter of a circle of equal area. 

The surface of a sphere equals the square of the diameter multi¬ 
plied by the constant 3.1416. 

The area of a triangle is equal to the base multiplied by one-half 
of the altitude. 

The area of a sector is equal to one-half of the length of the arc 
multiplied by the radius of the circle. 

To find the capacity (U. S. gallons) of cylindrical tanks, square the 
diameter expressed in inches, multiplying this product by the length, 
and then the last found product by the constant .0034. 

To find the number of square heating surface in tubes, multiply 
the number of tubes by the diameter of a tube in inches, and then this 
product the length of a tube in feet, and this last found product by the 
constant .2618. 








APR 15 1912 



















































































PUNTON-CLARK PRESS, KANSAS CITY, MO. 





















































































* • 






















ft 










\ 


























































































/ 



















































LIBRARY OF CONGRESS 



0 033 266 582*0 

































































































































































































































































































































































































































